https://baiku.ai/ daily 1 https://baiku.ai/courses daily 0.9 https://baiku.ai/courses/new monthly 0.5 https://baiku.ai/pricing monthly 0.7 https://baiku.ai/courses/space-101 weekly 0.6 https://baiku.ai/courses/how-your-body-works weekly 0.6 https://baiku.ai/courses/money-basics weekly 0.6 https://baiku.ai/courses/ai-foundations weekly 0.6 https://baiku.ai/courses/climate-essentials weekly 0.6 https://baiku.ai/https://en.wikipedia.org/wiki/UNESCO 2026-08-03T04:08:01.831Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Bubble_sort 2026-08-03T02:42:54.651Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Vaccine 2026-08-03T02:42:31.356Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/List_comprehension 2026-08-03T02:42:01.923Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/AI_relay_service 2026-08-03T02:36:18.784Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Relay_service 2026-08-03T02:35:21.687Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Python_(programming_language) 2026-08-03T01:58:49.600Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Jim_Corbett_National_Park 2026-08-03T01:22:31.463Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Deer 2026-08-02T23:06:39.536Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Tell_me_about_moose%E2%80%99s_horn 2026-08-02T22:57:58.449Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Chromosome 2026-08-02T18:31:14.589Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Environmental_science 2026-08-02T17:49:06.417Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Financial_institution 2026-08-02T17:46:27.341Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Molecular_biology 2026-08-02T17:42:19.501Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Tide 2026-08-02T17:39:03.462Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Cell_(biology) 2026-08-02T17:31:33.119Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Genetic_code 2026-08-02T17:20:13.096Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Newton's_laws_of_motion 2026-08-02T17:16:28.024Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Carbon_cycle 2026-08-02T16:52:06.504Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Moon 2026-08-02T16:25:13.594Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Meteoroid 2026-08-02T16:24:57.906Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Carbon_footprint 2026-08-02T16:12:09.149Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/NBDHE 2026-08-02T16:01:32.762Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Consumer_price_index 2026-08-02T15:59:52.477Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Central_dogma_of_molecular_biology 2026-08-02T14:54:21.491Z monthly 0.5 https://baiku.ai/https://fr.wikipedia.org/wiki/Soleil 2026-08-02T14:21:24.628Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Air_pollution 2026-08-02T14:12:29.273Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Fossil_fuel 2026-08-02T13:55:49.009Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Greenhouse_gas 2026-08-02T13:46:00.114Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Conservation_biology 2026-08-02T13:45:45.067Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Axolotl 2026-08-02T13:41:26.436Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Enzyme 2026-08-02T13:01:57.550Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/& 2026-08-02T12:59:58.307Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Evolution 2026-08-02T12:58:32.489Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Population_dynamics 2026-08-02T12:32:30.924Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Comet 2026-08-02T12:30:54.049Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Solar_System 2026-08-02T11:31:12.730Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Translation_(biology) 2026-08-02T11:28:04.689Z monthly 0.5 https://baiku.ai/https://es.wikipedia.org/wiki/Estrella 2026-08-02T09:24:34.822Z monthly 0.5 https://baiku.ai/https://www.kyoto-u.ac.jp/en/research-news/2026-07-28 2026-08-02T01:02:26.824Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Neutron_star 2026-08-01T15:34:32.923Z monthly 0.5 https://baiku.ai/https://acoup.blog/2026/01/30/collections-the-late-bronze-age-collapse-a-very-brief-introduction 2026-07-31T19:20:03.421Z monthly 0.5 https://baiku.ai/https://acoup.blog/2026/01/30/collections-the-late-bronze-age-collapse-a-very-brief-introduction/ 2026-07-31T19:20:00.785Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Human_body 2026-07-31T18:44:35.810Z monthly 0.5 https://baiku.ai/https://es.wikipedia.org/wiki/Cuerpo_humano 2026-07-31T18:43:32.340Z monthly 0.5 https://baiku.ai/https://es.wikipedia.org/wiki/Cuerpo_humano 2026-07-31T18:41:53.072Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Human_body 2026-07-31T18:40:56.881Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Chess 2026-07-31T18:40:13.732Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/volcanoes 2026-07-31T18:38:17.413Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Chess_strategy 2026-07-31T18:20:53.665Z monthly 0.5 https://baiku.ai/https://www.springerprofessional.de/en/turnpike-phenomenon-for-markov-decision-processes/51520638 2026-07-31T18:13:11.004Z monthly 0.5 https://baiku.ai/https://www.springerprofessional.de/en/springerbriefs-in-mathematics/1766134 2026-07-31T18:11:22.003Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Discussion%3A_In_a_question_like_this_it_is_as_important_to_realize_what_it_is_not_asking_as_what_it_is_asking._It_is_not_asking_you_to_determine_the_values_of_a_and_b._How_can_that_help%3F_Get_Your_Hands_Dirty_Practice_until_concepts_have_become_so_obvious%2C_so_intuitive%2C_that_you_could_handle_them_without_thinking_%E2%80%93_in_your_sleep._You_must_see_them_in_your_eye%2C_have_them_right_in_your_fingers._%E2%80%93_Benoit_Mandelbrot_Get_Your_Hands_Dirty%3A_Try_some_sample_computations._Do_some_experiments._Draw_some_pictures._Build_models._Play_with_the_%E2%80%9Ctoys%E2%80%9D_that_are_given_to_you_in_the_question._Muck_around._If_the_question_asks_you_to_prove_something_for_all_natural_numbers_n%2C_try_it_for_n_%3D_0%2C_1%2C_2%2C_3%2C_4%2C_5._Playing_and_computing_and_doing_sample_calculations_and_experimenting_can_build_insight_into_what_is_actually_going_on._If_you_are_very_lucky%2C_sometimes_a_few_sample_computations_are_all_that_is_needed_to_solve_the_problem._P_R_O_B_L_E_M_Suppose_12a_%2B_10b_%3D_1020._Find_a_5_%2B_b_6_._Discussion%3A_How_can_we_get_our_hands_dirty_in_such_a_problem%3F_Can_it_be_helpful%3F_P_R_O_B_L_E_M_Find_all_prime_numbers_that_are_the_sum_of_four_consecutive_prime_numbers._Discussion%3A_How_can_we_get_our_hands_dirty_in_this_problem%3F_Consider_the_Penultimate_Step_Consider_the_Penultimate_Step%3A_It_is_often_helpful_to_consider_what_the_next_to_last_step_in_the_solution_could_be_in_order_to_solve_the_question._This_is_%E2%80%9Cworking%E2%80%9D_backwards_from_the_desired_goal._This_can_be_generalized_by_considering_the_step_before_the_penultimate_%C2%A9_2022_KEN_MONKS_PAGE_6_of_23_The_Art_of_Problem_Solving_step%2C_and_so_on%2C_working_backwards_from_the_goal_and_forwards_from_the_hypotheses_in_the_hope_of_meeting_up_somewhere_in_the_middle._Common_Pitfalls%3A_Note_that_often_there_is_more_than_one_penultimate_step_possible%2C_and_you_should_remain_open_to_all_possibilities_rather_than_committing_yourself_to_the_first_plan_of_attack_that_comes_to_mind%2C_which_may_inevitably_prove_to_be_impossible_or_unwieldy._P_R_O_B_L_E_M_In_triangle_%E2%96%B3ABC%2C_point_D_on_BC_is_equidistant_from_the_vertices_A%2C_B%2C_and_C._Prove_that_%7CAB%7C_2_%2B_%7CAC%7C_2_%3D_%7CBC%7C_2_._Discussion%3A_Where_have_we_seen_this_kind_of_equation_before%3F_What_penultimate_step_would_suffice_to_prove_such_an_equation%3F_Consider_a_Simpler_Problem_If_the_given_problem_is_too_hard%2C_solve_an_easier_one._%E2%80%93_Zeitz_Consider_a_Simpler_Problem%3A_Another_way_to_gain_an_insight_into_a_difficult_problem_is_to_try_solving_a_simpler_problem_that_is_similar_to_the_difficult_one._This_may_involve_solving_the_same_problem_with_fewer_variables%2C_or_smaller_numbers._P_R_O_B_L_E_M_How_many_ordered_triples_of_positive_integers_sum_to_20%3F_Discussion%3A_What_is_problems_can_you_think_of_that_are_similar_to_this_one%2C_but_seem_to_be_simpler%3F_Can_solving_the_simpler_problem_help%3F_%C2%A9_2022_KEN_MONKS_PAGE_7_of_23_The_Art_of_Problem_Solving_Wishful_Thinking_Good%2C_obedient_boys_and_girls_solve_fewer_problems_than_naughty_and_mischievous_ones._%E2%80%93_Zeitz_Wishful_Thinking%3A_It_is_sometimes_helpful_to_consider_something_that_is_blatantly_false_that_you_wish_were_true_because_it_would_make_the_problem_much_easier_to_solve._In_addition_to_giving_some_glimpses_of_how_the_problem_might_be_solved%2C_understanding_why_the_thing_you_wish_to_be_true_is_false_is_often_essential_to_understanding_the_key_difficulty_in_the_problem._P_R_O_B_L_E_M_The_product_of_five_consecutive_integers_is_2441880._What_is_the_largest_of_the_five_integers%3F_Discussion%3A_How_can_we_attack_this_with_wishful_thinking%3F_Can_you_solve_it_cleverly_without_a_calculator_and_with_the_minimum_amount_of_arithmetic_(i.e.%2C_without_dumbassing_it)%3F_Common_Strategies_for_Proofs_If_it_is_a_Miracle%2C_any_sort_of_evidence_will_answer%2C_but_if_it_is_a_Fact%2C_proof_is_necessary._%E2%80%93_Mark_Twain_Informally%2C_a_proof_is_just_an_explanation_that_logically_guarantees_the_correctness_of_a_statement._More_formally%2C_a_proof_is_a_sequence_of_statements_that_are_either_(a)_a_definition_(b)_a_previously_proven_statement_or_(c)_follow_from_previous_statements_in_the_proof_by_logical_rules_of_inference._A_simple_example_of_formal_proofs_are_the_statement-reason_proofs_that_you_did_in_high_school_geometry._In_a_problem_solving%2C_most_proofs_are_not_written_in_that_statement-reason_two_column_format_but_rather_are_written_informally_as_careful_detailed_explanation_that_leads_the_reader_inexorably_to_the_desired_conclusion_with_no_ambiguity._But_even_though_the_style_%C2%A9_2022_KEN_MONKS_PAGE_8_of_23_The_Art_of_Problem_Solving_of_informally_written_proofs_differs_from_that_of_a_formal_proof%2C_the_content_is_exactly_the_same%3A_every_claim_in_your_proof_must_follow_from_previous_statements_by_the_rules_of_logic_or_must_be_a_previously_proven_result%2C_definition%2C_or_a_fact_given_in_the_statement_of_the_problem._Thus%2C_you_should_strive_to_give_reasons_for_every_claim_you_make_in_a_proof_and_show_why_it_follows_from_what_you_have_said_previously._There_are_several_very_common_arguments_used_in_proofs_that_every_problem_solver_must_be_familiar_with._1._Proving_a_Conditional_statement%3A_To_prove_a_statement_of_the_form_%22If_P_then_Q%22_where_P_and_Q_are_statements%2C_you_should_assume_P_is_true_and_then_prove_that_Q_is_true_under_that_assumption._2._Proving_a_Contrapositive%3A_To_prove_a_conditional_statement_like_%22If_P_then_Q%22_it_is_sometimes_easier_to_prove_the_equivalent_statement_%22If_not_Q_then_not_P%22_by_assuming_that_Q_is_false_and_showing_that_P_is_false_under_that_assumption._3._Proof_by_Contradiction%3A_To_prove_a_statement_P_sometime_it_is_useful_to_assume_P_is_false_and_then_show_that_that_assumption_leads_to_a_contradiction%2C_i.e.%2C_that_you_can_prove_both_some_statement_and_its_negation._4._Proof_by_Cases%3A_If_you_know_that_either_P_or_Q_is_true%2C_and_want_to_show_R%2C_you_can_prove_R_by_considering_separate_cases._In_the_first_case%2C_assume_P_and_prove_R._In_the_second_case_assume_Q_and_show_R._Since_one_of_P_or_Q_is_known_to_be_true%2C_R_must_be_true_as_well._This_method_also_generalizes_to_situations_where_you_have_more_than_two_cases_to_consider._5._Proof_of_Univerality%3A_To_prove_that_a_statement_is_true_about_every_element_in_some_set%2C_let_x_be_an_arbitrary%2C_unspecified_element_of_that_set_and_prove_that_the_statement_is_true_about_x._6._Proof_of_Existence%3A_To_prove_that_a_mathematical_object_with_certain_properties_exists%2C_either_make_an_example_of_such_an_object_(i.e.%2C_construct_one)%2C_or_show_that_if_it_didn%E2%80%99t_exist_there_would_be_a_contradiction_(i.e.%2C_use_proof_by_contradiction)._7._Proof_by_Induction%3A_Let_P_(n)_be_a_statement_about_an_unspecified_natural_number_n._To_prove_that_P_is_true_for_all_natural_number_values_of_n%2C_show_P_(0)_is_true%2C_then_let_k_be_a_natural_number_and_assuming_P_(k)_is_true%2C_show_that_P_(k_%2B_1)_is_also_true._P_R_O_B_L_E_M_The_plane_is_divided_into_regions_by_finitely_many_straight_lines._Show_that_it_is_always_possible_to_color_the_regions_with_two_colors_so_that_adjacent_regions_are_never_the_same_color_(like_a_checkerboard)._%C2%A9_2022_KEN_MONKS_PAGE_9_of_23_The_Art_of_Problem_Solving_4_Tactics_Zeitz_identifies_the_following_tactics%2C_which_apply_to_a_wide_range_of_problems._Symmetry_Look_for_harmony_and_beauty%2C_whenever_you_investigate_a_problem._If_you_can_do_something_that_makes_things_more_harmonious_or_more_beautiful%2C_even_if_you_have_no_idea_how_to_define_these_two_terms%2C_then_you_are_often_on_the_right_track._%E2%80%93_Zeitz_Tactic%3A_When_a_problem_has_symmetry%2C_try_to_use_it._Try_to_maintain_that_symmetry_while_solving_the_problem._If_a_problem_doesn%E2%80%99t_have_symmetry%2C_but_you_wish_it_did%2C_try_to_introduce_symmetry_into_the_situation_if_possible._A_mathematical_object_(shape%2C_expression%2C_system_of_equations%2C_etc.)_is_symmetric_with_respect_to_some_action_or_operation_if_it_is_unchanged_by_the_action_or_operation._The_actions_that_do_this_are_called_the_symmetries_of_the_object._P_R_O_B_L_E_M_(MATHCOUNTS_2002_Workout_8%2C_number_8)_The_points_A%2C_B_and_C_lie_in_a_plane_and_have_coordinates_(6%2C_5)%2C_(2%2C_1)_and_(0%2C_k)%2C_respectively._What_value_of_k_makes_the_sum_of_the_lengths_of_segments_AC_and_BC_the_least_possible_value%3F_Examples_of_Symmetry_Groups_A_group_is_a_set_together_with_an_associative_binary_operator_on_that_set_that_has_an_identity_element_and_inverses_for_every_element_in_the_set._The_set_of_symmetries_of_an_object_often_forms_a_group._The_set_%7B1%2C_2%2C_3%2C_._._._%2C_n%7D_is_unchanged_by_permuting_its_elements._The_set_of_all_permutations_of_%7B1%2C_2%2C_3%2C_._._._%2C_n%7D_is_called_the_symmetric_group_Sn. 2026-07-31T18:09:52.567Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Discussion:_In_a_question_like_this_it_is_as_important_to_realize_what_it_is_not_asking_as_what_it_is_asking._It_is_not_asking_you_to_determine_the_values_of_a_and_b._How_can_that_help?_Get_Your_Hands_Dirty_Practice_until_concepts_have_become_so_obvious,_so_intuitive,_that_you_could_handle_them_without_thinking_%E2%80%93_in_your_sleep._You_must_see_them_in_your_eye,_have_them_right_in_your_fingers._%E2%80%93_Benoit_Mandelbrot_Get_Your_Hands_Dirty:_Try_some_sample_computations._Do_some_experiments._Draw_some_pictures._Build_models._Play_with_the_%E2%80%9Ctoys%E2%80%9D_that_are_given_to_you_in_the_question._Muck_around._If_the_question_asks_you_to_prove_something_for_all_natural_numbers_n,_try_it_for_n_=_0,_1,_2,_3,_4,_5._Playing_and_computing_and_doing_sample_calculations_and_experimenting_can_build_insight_into_what_is_actually_going_on._If_you_are_very_lucky,_sometimes_a_few_sample_computations_are_all_that_is_needed_to_solve_the_problem._P_R_O_B_L_E_M_Suppose_12a_+_10b_=_1020._Find_a_5_+_b_6_._Discussion:_How_can_we_get_our_hands_dirty_in_such_a_problem?_Can_it_be_helpful?_P_R_O_B_L_E_M_Find_all_prime_numbers_that_are_the_sum_of_four_consecutive_prime_numbers._Discussion:_How_can_we_get_our_hands_dirty_in_this_problem?_Consider_the_Penultimate_Step_Consider_the_Penultimate_Step:_It_is_often_helpful_to_consider_what_the_next_to_last_step_in_the_solution_could_be_in_order_to_solve_the_question._This_is_%E2%80%9Cworking%E2%80%9D_backwards_from_the_desired_goal._This_can_be_generalized_by_considering_the_step_before_the_penultimate_%C2%A9_2022_KEN_MONKS_PAGE_6_of_23_The_Art_of_Problem_Solving_step,_and_so_on,_working_backwards_from_the_goal_and_forwards_from_the_hypotheses_in_the_hope_of_meeting_up_somewhere_in_the_middle._Common_Pitfalls:_Note_that_often_there_is_more_than_one_penultimate_step_possible,_and_you_should_remain_open_to_all_possibilities_rather_than_committing_yourself_to_the_first_plan_of_attack_that_comes_to_mind,_which_may_inevitably_prove_to_be_impossible_or_unwieldy._P_R_O_B_L_E_M_In_triangle_%E2%96%B3ABC,_point_D_on_BC_is_equidistant_from_the_vertices_A,_B,_and_C._Prove_that_|AB|_2_+_|AC|_2_=_|BC|_2_._Discussion:_Where_have_we_seen_this_kind_of_equation_before?_What_penultimate_step_would_suffice_to_prove_such_an_equation?_Consider_a_Simpler_Problem_If_the_given_problem_is_too_hard,_solve_an_easier_one._%E2%80%93_Zeitz_Consider_a_Simpler_Problem:_Another_way_to_gain_an_insight_into_a_difficult_problem_is_to_try_solving_a_simpler_problem_that_is_similar_to_the_difficult_one._This_may_involve_solving_the_same_problem_with_fewer_variables,_or_smaller_numbers._P_R_O_B_L_E_M_How_many_ordered_triples_of_positive_integers_sum_to_20?_Discussion:_What_is_problems_can_you_think_of_that_are_similar_to_this_one,_but_seem_to_be_simpler?_Can_solving_the_simpler_problem_help? 2026-07-31T18:09:08.787Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Discussion%3A_In_a_question_like_this_it_is_as_important_to_realize_what_it_is_not_asking_as_what_it_is_asking._It_is_not_asking_you_to_determine_the_values_of_a_and_b._How_can_that_help%3F_Get_Your_Hands_Dirty_Practice_until_concepts_have_become_so_obvious%2C_so_intuitive%2C_that_you_could_handle_them_without_thinking_%E2%80%93_in_your_sleep._You_must_see_them_in_your_eye%2C_have_them_right_in_your_fingers._%E2%80%93_Benoit_Mandelbrot_Get_Your_Hands_Dirty%3A_Try_some_sample_computations._Do_some_experiments._Draw_some_pictures._Build_models._Play_with_the_%E2%80%9Ctoys%E2%80%9D_that_are_given_to_you_in_the_question._Muck_around._If_the_question_asks_you_to_prove_something_for_all_natural_numbers_n%2C_try_it_for_n_%3D_0%2C_1%2C_2%2C_3%2C_4%2C_5._Playing_and_computing_and_doing_sample_calculations_and_experimenting_can_build_insight_into_what_is_actually_going_on._If_you_are_very_lucky%2C_sometimes_a_few_sample_computations_are_all_that_is_needed_to_solve_the_problem._P_R_O_B_L_E_M_Suppose_12a_%2B_10b_%3D_1020._Find_a_5_%2B_b_6_._Discussion%3A_How_can_we_get_our_hands_dirty_in_such_a_problem%3F_Can_it_be_helpful%3F_P_R_O_B_L_E_M_Find_all_prime_numbers_that_are_the_sum_of_four_consecutive_prime_numbers._Discussion%3A_How_can_we_get_our_hands_dirty_in_this_problem%3F_Consider_the_Penultimate_Step_Consider_the_Penultimate_Step%3A_It_is_often_helpful_to_consider_what_the_next_to_last_step_in_the_solution_could_be_in_order_to_solve_the_question._This_is_%E2%80%9Cworking%E2%80%9D_backwards_from_the_desired_goal._This_can_be_generalized_by_considering_the_step_before_the_penultimate_%C2%A9_2022_KEN_MONKS_PAGE_6_of_23_The_Art_of_Problem_Solving_step%2C_and_so_on%2C_working_backwards_from_the_goal_and_forwards_from_the_hypotheses_in_the_hope_of_meeting_up_somewhere_in_the_middle._Common_Pitfalls%3A_Note_that_often_there_is_more_than_one_penultimate_step_possible%2C_and_you_should_remain_open_to_all_possibilities_rather_than_committing_yourself_to_the_first_plan_of_attack_that_comes_to_mind%2C_which_may_inevitably_prove_to_be_impossible_or_unwieldy._P_R_O_B_L_E_M_In_triangle_%E2%96%B3ABC%2C_point_D_on_BC_is_equidistant_from_the_vertices_A%2C_B%2C_and_C._Prove_that_%7CAB%7C_2_%2B_%7CAC%7C_2_%3D_%7CBC%7C_2_._Discussion%3A_Where_have_we_seen_this_kind_of_equation_before%3F_What_penultimate_step_would_suffice_to_prove_such_an_equation%3F_Consider_a_Simpler_Problem_If_the_given_problem_is_too_hard%2C_solve_an_easier_one._%E2%80%93_Zeitz_Consider_a_Simpler_Problem%3A_Another_way_to_gain_an_insight_into_a_difficult_problem_is_to_try_solving_a_simpler_problem_that_is_similar_to_the_difficult_one._This_may_involve_solving_the_same_problem_with_fewer_variables%2C_or_smaller_numbers._P_R_O_B_L_E_M_How_many_ordered_triples_of_positive_integers_sum_to_20%3F_Discussion%3A_What_is_problems_can_you_think_of_that_are_similar_to_this_one%2C_but_seem_to_be_simpler%3F_Can_solving_the_simpler_problem_help%3F 2026-07-31T18:09:06.680Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/The_Art_of_Problem_Solving_Math_479_-_Dr._Monks_Contemplation_within_activity_is_a_million_times_better_than_contemplation_within_stillness._%E2%80%93_Hakuin_1_The_Way_of_Problem_Solving_%E2%80%A2_Art%3A_Problem_solving_is_an_art._Like_any_art_it_requires_proper_attitude%2C_practice%2C_creativity%2C_and_passion_to_master._Like_any_artist_the_problem_solver_creates_works_of_wonder_and_surprise_and_sublime_aesthetic_value._%E2%80%A2_Beauty%3A_A_correct_solution_is_better_than_no_solution._A_massive_straightforward_slog_(a.k.a._dumb-assing)_that_gets_the_correct_answer_is_better_than_no_solution_at_all._But_a_clever_correct_solution_is_better_than_a_straightforward_or_obvious_solution._All_else_being_equal%2C_the_shorter_the_solution%2C_the_better._A_solution_that_does_not_require_a_calculator_or_computer_is_better_than_one_that_does._A_solution_that_does_not_require_algebra_is_better_than_one_that_does._%E2%80%A2_Fellowship%3A_As_with_any_art_form%2C_we_can_benefit_from_interacting_with_other_artists._By_aspiring_to_learn_from_those_who_are_more_experienced%2C_by_cooperating_with_our_peers%2C_and_by_assisting_those_who_are_less_experienced%2C_everyone_benefits._Several_minds_can_produce_several_perspectives_on_the_same_problem._As_with_any_group_of_artisans%2C_problem_solvers_naturally_bond_together_into_a_community_of_people_who_share_a_common_interest._The_life_of_Zen_attainment_is_not_like_standing_on_a_riverbank_watching_the_current_and_appreciating_the_water_or_landscape_as_a_witness%3B_it_is_jumping_into_the_current_and_becoming_one_with_it._%E2%80%93_Leggett_%C2%A9_2022_KEN_MONKS_PAGE_1_of_23_The_Art_of_Problem_Solving_1.1_Problems_vs_Exercises_Zeitz_distinguishes_between_a_problem_and_an_exercise._Some_Characteristics_of_a_Good_Problem_%E2%80%A2_Your_first_impression_is_%E2%80%9CThis_is_impossible!%E2%80%9D._%E2%80%A2_You_are_surprised%2C_or_delighted_by_the_question_itself._%E2%80%A2_It_is_simple_to_state%2C_but_hard_to_answer._%E2%80%A2_You%E2%80%99ve_never_seen_a_question_just_like_it_before._%E2%80%A2_You_don%E2%80%99t_know_immediately_how_to_solve_it._%E2%80%A2_It_is_addictive._You_want_to_know_the_answer._%E2%80%A2_It_has_an_obvious%2C_straightforward%2C_ugly%2C_messy%2C_lengthy_solution%2C_but_also_a_clever%2C_ingenious%2C_short%2C_elegant_solution._%E2%80%A2_It_has_some_symmetry%2C_or_a_story%2C_or_a_pattern_or_picture_that_is_aesthetically_pleasing._%E2%80%A2_It_only_requires_only_very_elementary_mathematics%2C_but_is_quite_challenging_nonetheless._1.2_Some_Problem_Solving_Terminology_and_Folklore_%E2%80%A2_Crux_move%3A_Zeitz_calls_a_key_insight_or_key_step_in_the_solution_to_a_problem_a_crux_move._It_is_the_key_realization_that_allows_you_to_solve_the_problem%2C_all_other_parts_of_the_solution_being_more_or_less_straightforward._A_problem_can_have_more_than_one_crux_move._It_refers_to_the_most_difficult%2C_tricky%2C_or_creative_step_or_steps_in_a_solution._%E2%80%A2_Fence_post_error%3A_Probably_the_number_one_killer_of_otherwise_perfectly_good_solutions_is_being_off_by_one_when_counting_something._This_gets_its_name_from_a_problem_similar_to_this_one._A_fence_is_constructed_of_ten_foot_sections_consisting_of_two_horizontal_bars_supported_by_posts_at_each_end._How_many_fence_posts_are_needed_to_construct_a_straight_fence_of_this_type_that_is_100_ft_long%3F_If_your_answer_is_ten%2C_then_you_have_committed_the_fence_post_error._We_say_a_proposed_solution_is_%E2%80%9Coff_by_a_fence_post%E2%80%9D_to_mean_that_the_solution_is_wrong_because_something_was_counted_incorrectly_and_the_count_came_out_to_be_either_one_to_high_or_one_two_low._As_you_solve_many_problems_you_will_learn_to_have_a_healthy_hatred_of_fence_posts._%E2%80%A2_Pwn%3A_refers_to_dominating_of_an_opponent%2C_or_something_great_or_ingenious_applied_to_methods_or_objects._This_term_is_used_by_problem_solvers_to_indicate_a_great_ingenious_solution_to_a_problem%2C_one_that_dominates_and_completely_solves_the_problem_in_the_best_possible_manner._%E2%80%A2_Spoilers%3A_it_is_traditional_in_problem_solving%2C_just_as_with_movies_or_puzzles%2C_to_respect_other%E2%80%99s_rights_and_desire_to_enjoy_a_problem_to_its_fullest._Therefore_it_is_%C2%A9_2022_KEN_MONKS_PAGE_2_of_23_The_Art_of_Problem_Solving_important_not_to_provide_a_%E2%80%9Cspoiler%E2%80%9D_to_someone_who_is_working_hard_on_a_problem_by_revealing_the_solution_or_giving_them_a_big_hint_unless_they_ask_you_for_it_first._Also_it_is_much_more_rewarding_to_solve_a_problem_if_you_only_have_a_little_hint_than_if_someone_just_tells_you_the_solution._A_good_problem_solver_respects_their_fellow_problem_solvers_need_to_enjoy_the_solution_and_will_devote_some_thought_to_giving_%E2%80%9Cjust_the_right_hint%E2%80%9D_if_someone_requests_it_so_as_not_to_spoil_their_fun._1.3_Why_Problem_Solving%3F_%E2%80%A2_For_Pure_Mathematicians%3A_the_closest_activity_in_mathematics_to_problem_solving_is_mathematical_research._In_both_research_and_problem_solving_the_mathematician_must_learn_how_to_solve_problems_whose_solution_is_not_immediately_apparent._All_of_the_same_strategies%2C_tactics%2C_tools%2C_skills%2C_and_attitudes_that_are_used_by_the_problem_solver_can_also_be_used_effectively_by_the_researcher._Problem_solving_provides_an_excellent_training_ground_for_research_in_a_more_well-defined_environment_in_which_the_problems_are_known_to_have_a_solution_and_are_meant_to_be_solved_in_only_a_few_hours_rather_than_over_several_months_or_years._%E2%80%A2_For_Applied_Mathematicians%3A_the_applied_mathematician_or_scientist_benefits_from_a_problem_solving_background_by_practicing_to_be_accurate%2C_careful%2C_creative%2C_and_confident_when_faced_with_a_problem._The_applied_mathematician_is_often_faced_with_a_complicated_or_ill_defined_problem_and_must_use_his_skills_as_a_problem_solver_to_come_to_a_deeper_understanding_of_the_problem_and_solve_it._Just_as_in_problem_solving%2C_an_efficient_solution_to_a_problem_is_often_substantially_more_valuable_than_an_inefficient_one._%E2%80%A2_For_Math_Teachers%3A_problem_solving_has_its_roots_in_mathematics_competitions_starting_at_the_4th_grade_level_and_continuing_on_up_through_the_Putnam_exam._Mathematics_teachers_who_have_a_problem_solving_background_gain_a_substantially_deeper_understanding_of_the_topics_they_must_teach_and_gain_the_ability_to_coach_students_who_wish_to_participate_in_mathematical_competitions._1.4_The_Problem_Solving_Mindset_If_you_would_be_a_real_seeker_after_truth%2C_it_is_necessary_that_at_least_once_in_your_life_you_doubt%2C_as_far_as_possible%2C_all_things._%E2%80%93_Descartes_There_are_several_attitudes_or_psychological_perspectives_that_are_needed_to_be_a_successful_problem_solver._%E2%80%A2_Concentration%3A_it_is_easy_to_get_distracted_or_frustrated_by_a_difficult_problem._Problem_solving_requires_sometimes_lengthy%2C_intense%2C_focused_concentration_on_a_single_topic._%C2%A9_2022_KEN_MONKS_PAGE_3_of_23_The_Art_of_Problem_Solving_%E2%80%A2_Confidence%3A_it_is_important_to_believe_that_you_will_eventually_be_able_to_solve_a_problem%2C_even_if_you_have_no_idea_how_to_do_it_at_first._Even_if_you_are_a_beginner_at_problem_solving%2C_you_should_approach_a_problem_with_a_confident_attitude._Don%E2%80%99t_worry_that_you_might_not_remember_a_key_theorem_or_an_important_fact._Every_problem_has_to_be_solved_with_what_you_already_know._%E2%80%A2_Creativity%3A_a_problem_solver_must_always_remain_open_to_all_and_any_ideas_that_may_come_to_mind_and_always_on_the_lookout_for_new_ways_to_approach_a_problem._A_change_of_perspective%2C_a_reinterpretation_of_the_question%2C_a_nonstandard_approach_to_a_otherwise_familiar_situation_can_have_tremendous_benefits._It_can_also_be_a_dead_end._But_if_even_one_idea_in_ten_is_fruitful%2C_that_may_be_the_only_one_you_need_to_solve_the_problem._%E2%80%A2_Peripheral_vision%3A_when_looking_at_the_night_sky_we_can_see_fainter_objects_by_not_looking_directly_at_them._The_receptors_on_the_sides_of_our_eyes_are_more_sensitive_to_faint_light_than_those_in_the_center._Similarly%2C_when_solving_a_problem%2C_we_should_not_always_think_about_solving_the_problem_itself_directly%2C_but_rather_allow_ourselves_to_ponder_things_that_are_perhaps_only_vaguely_related_to_the_problem._This_is_similar_to_a_smell_or_gut_instinct_or_intuition_that_leads_you_in_a_certain_direction_without_being_100%25_certain_why_you_think_you_ought_to_go_that_way._The_more_you_practice%2C_the_more_reliable_your_instincts_will_become._%E2%80%A2_Thinking_on_your_feet%3A_problem_solvers_strive_to_develop_the_ability_to_think_on_their_feet_with_the_minimal_amount_of_assistance_possible._A_solution_that_does_not_require_a_calculator_or_computer_is_better_than_one_that_does._A_solution_that_does_not_require_a_pencil_and_paper_is_better_than_one_that_does._A_short_elementary_solution_that_does_not_require_any_advanced_theorems_or_previously_proven_results_is_better_than_one_that_does._The_problem_solver_solves_problems_in_the_shower%2C_while_lying_in_bed_before_going_to_sleep_or_right_after_waking_up%2C_while_running_or_biking_or_hiking_or_driving_in_the_car._The_problem_solver_may_actually_look_forward_to_time_in_the_waiting_room_at_the_doctor%E2%80%99s_office_or_dentist_as_it_provides_uninterrupted_time_to_work_on_their_problems._%E2%80%A2_Stay_loose%3A_The_mind_is_a_more_flexible_and_fluid_canvass_than_pencil_and_blank_paper._We_can_manipulate_ideas_freely_in_our_mind._Putting_something_down_on_paper_tends_to_make_it_more_concrete_and_cast_in_stone._The_more_you_practice%2C_the_better_you_will_become_at_not_needing_paper_and_pencil_to_do_mathematics._As_you_do_you_will_sometimes_find_that_you_have_more_success_solving_difficult_problems_if_you_don%E2%80%99t_use_paper_than_if_you_do!_Especially_at_the_beginning%2C_when_you_first_approach_a_problem%2C_it_is_important_to_stay_loose_and_flexible._Working_mostly_in_your_head_is_often_the_best_way_to_do_that._Once_you_have_an_epiphany_and_see_the_crux_move%2C_it_may_then_be_time_to_break_out_the_paper_or_calculators._%E2%80%A2_Be_careful%3A_without_accuracy_and_care%2C_stupid_mistakes_can_easily_turn_an_otherwise_correct_solution_into_an_incorrect_one._Also_some_problems_may_be_easy_to_solve_if_you_do_them_correctly_but_a_hideous_nightmare_if_you_make_a_small_mistake._The_%C2%A9_2022_KEN_MONKS_PAGE_4_of_23_The_Art_of_Problem_Solving_problem_solver_must_also_strive_to_be_sure_that_every_case_has_been_considered_and_that_there_is_no_omission_in_the_solution_that_could_catastrophic._2_Master_Zeitz%E2%80%99s_Threefold_Path_If_you_do_not_follow_the_right_path%2C_you_will_be_lost._%E2%80%93_The_Buddha_The_experienced_problem_solver_operates_on_three_different_levels%3A_1._Strategy%3A_mathematical_and_psychological_ideas_for_starting_and_pursuing_problems._2._Tactics%3A_Diverse_mathematical_methods_that_work_in_many_different_settings._3._Tools%3A_Narrowly_focused_techniques_and_%E2%80%9Ctricks%E2%80%9D_for_specific_situations._3_Strategies_Zeitz_identifies_the_following_strategies._Get_Oriented_Read_the_problem!_%E2%80%93_Monks_Get_Oriented%3A_Take_time_to_understand_exactly_what_the_question_is_asking._Notice_every_word_and_make_a_mental_inventory_of_everything_you_are_given%2C_and_exactly_what_you_are_asked._Words_and_phrases_like_%E2%80%9Cpositive%E2%80%9D_or_%E2%80%9Cat_most%E2%80%9D_or_%E2%80%9Cinteger%E2%80%9D_or_%E2%80%9Cunique%E2%80%9D_can_be_crucial._Also_be_aware_of_what_the_question_does_not_say._Don%E2%80%99t_assume_anything_that_isn%E2%80%99t_stated_in_the_question_and_don%E2%80%99t_ignore_anything_that_is._Common_pitfalls%3A_It_is_very_easy_to_interpret_a_question_the_wrong_way_by_skipping_a_single_word%2C_or_incorrectly_identifying_it_as_another_similar_question_that_you_are_more_familiar_with._P_R_O_B_L_E_M_Suppose_12a_%2B_10b_%3D_1020._Find_a_5_%2B_b_6_._%C2%A9_2022_KEN_MONKS_PAGE_5_of_23_The_Art_of_Problem_Solving_Discussion%3A_In_a_question_like_this_it_is_as_important_to_realize_what_it_is_not_asking_as_what_it_is_asking._It_is_not_asking_you_to_determine_the_values_of_a_and_b._How_can_that_help%3F_Get_Your_Hands_Dirty_Practice_until_concepts_have_become_so_obvious%2C_so_intuitive%2C_that_you_could_handle_them_without_thinking_%E2%80%93_in_your_sleep._You_must_see_them_in_your_eye%2C_have_them_right_in_your_fingers._%E2%80%93_Benoit_Mandelbrot_Get_Your_Hands_Dirty%3A_Try_some_sample_computations._Do_some_experiments._Draw_some_pictures._Build_models._Play_with_the_%E2%80%9Ctoys%E2%80%9D_that_are_given_to_you_in_the_question._Muck_around._If_the_question_asks_you_to_prove_something_for_all_natural_numbers_n%2C_try_it_for_n_%3D_0%2C_1%2C_2%2C_3%2C_4%2C_5._Playing_and_computing_and_doing_sample_calculations_and_experimenting_can_build_insight_into_what_is_actually_going_on._If_you_are_very_lucky%2C_sometimes_a_few_sample_computations_are_all_that_is_needed_to_solve_the_problem._P_R_O_B_L_E_M_Suppose_12a_%2B_10b_%3D_1020._Find_a_5_%2B_b_6_._Discussion%3A_How_can_we_get_our_hands_dirty_in_such_a_problem%3F_Can_it_be_helpful%3F_P_R_O_B_L_E_M_Find_all_prime_numbers_that_are_the_sum_of_four_consecutive_prime_numbers._Discussion%3A_How_can_we_get_our_hands_dirty_in_this_problem%3F_Consider_the_Penultimate_Step_Consider_the_Penultimate_Step%3A_It_is_often_helpful_to_consider_what_the_next_to_last_step_in_the_solution_could_be_in_order_to_solve_the_question._This_is_%E2%80%9Cworking%E2%80%9D_backwards_from_the_desired_goal._This_can_be_generalized_by_considering_the_step_before_the_penultimate_%C2%A9_2022_KEN_MONKS_PAGE_6_of_23_The_Art_of_Problem_Solving_step%2C_and_so_on%2C_working_backwards_from_the_goal_and_forwards_from_the_hypotheses_in_the_hope_of_meeting_up_somewhere_in_the_middle._Common_Pitfalls%3A_Note_that_often_there_is_more_than_one_penultimate_step_possible%2C_and_you_should_remain_open_to_all_possibilities_rather_than_committing_yourself_to_the_first_plan_of_attack_that_comes_to_mind%2C_which_may_inevitably_prove_to_be_impossible_or_unwieldy._P_R_O_B_L_E_M_In_triangle_%E2%96%B3ABC%2C_point_D_on_BC_is_equidistant_from_the_vertices_A%2C_B%2C_and_C._Prove_that_%7CAB%7C_2_%2B_%7CAC%7C_2_%3D_%7CBC%7C_2_._Discussion%3A_Where_have_we_seen_this_kind_of_equation_before%3F_What_penultimate_step_would_suffice_to_prove_such_an_equation%3F_Consider_a_Simpler_Problem_If_the_given_problem_is_too_hard%2C_solve_an_easier_one._%E2%80%93_Zeitz_Consider_a_Simpler_Problem%3A_Another_way_to_gain_an_insight_into_a_difficult_problem_is_to_try_solving_a_simpler_problem_that_is_similar_to_the_difficult_one._This_may_involve_solving_the_same_problem_with_fewer_variables%2C_or_smaller_numbers._P_R_O_B_L_E_M_How_many_ordered_triples_of_positive_integers_sum_to_20%3F_Discussion%3A_What_is_problems_can_you_think_of_that_are_similar_to_this_one%2C_but_seem_to_be_simpler%3F_Can_solving_the_simpler_problem_help%3F_%C2%A9_2022_KEN_MONKS_PAGE_7_of_23_The_Art_of_Problem_Solving_Wishful_Thinking_Good%2C_obedient_boys_and_girls_solve_fewer_problems_than_naughty_and_mischievous_ones._%E2%80%93_Zeitz_Wishful_Thinking%3A_It_is_sometimes_helpful_to_consider_something_that_is_blatantly_false_that_you_wish_were_true_because_it_would_make_the_problem_much_easier_to_solve._In_addition_to_giving_some_glimpses_of_how_the_problem_might_be_solved%2C_understanding_why_the_thing_you_wish_to_be_true_is_false_is_often_essential_to_understanding_the_key_difficulty_in_the_problem._P_R_O_B_L_E_M_The_product_of_five_consecutive_integers_is_2441880._What_is_the_largest_of_the_five_integers%3F_Discussion%3A_How_can_we_attack_this_with_wishful_thinking%3F_Can_you_solve_it_cleverly_without_a_calculator_and_with_the_minimum_amount_of_arithmetic_(i.e.%2C_without_dumbassing_it)%3F_Common_Strategies_for_Proofs_If_it_is_a_Miracle%2C_any_sort_of_evidence_will_answer%2C_but_if_it_is_a_Fact%2C_proof_is_necessary._%E2%80%93_Mark_Twain_Informally%2C_a_proof_is_just_an_explanation_that_logically_guarantees_the_correctness_of_a_statement._More_formally%2C_a_proof_is_a_sequence_of_statements_that_are_either_(a)_a_definition_(b)_a_previously_proven_statement_or_(c)_follow_from_previous_statements_in_the_proof_by_logical_rules_of_inference._A_simple_example_of_formal_proofs_are_the_statement-reason_proofs_that_you_did_in_high_school_geometry._In_a_problem_solving%2C_most_proofs_are_not_written_in_that_statement-reason_two_column_format_but_rather_are_written_informally_as_careful_detailed_explanation_that_leads_the_reader_inexorably_to_the_desired_conclusion_with_no_ambiguity._But_even_though_the_style_%C2%A9_2022_KEN_MONKS_PAGE_8_of_23_The_Art_of_Problem_Solving_of_informally_written_proofs_differs_from_that_of_a_formal_proof%2C_the_content_is_exactly_the_same%3A_every_claim_in_your_proof_must_follow_from_previous_statements_by_the_rules_of_logic_or_must_be_a_previously_proven_result%2C_definition%2C_or_a_fact_given_in_the_statement_of_the_problem._Thus%2C_you_should_strive_to_give_reasons_for_every_claim_you_make_in_a_proof_and_show_why_it_follows_from_what_you_have_said_previously._There_are_several_very_common_arguments_used_in_proofs_that_every_problem_solver_must_be_familiar_with._1._Proving_a_Conditional_statement%3A_To_prove_a_statement_of_the_form_%22If_P_then_Q%22_where_P_and_Q_are_statements%2C_you_should_assume_P_is_true_and_then_prove_that_Q_is_true_under_that_assumption._2._Proving_a_Contrapositive%3A_To_prove_a_conditional_statement_like_%22If_P_then_Q%22_it_is_sometimes_easier_to_prove_the_equivalent_statement_%22If_not_Q_then_not_P%22_by_assuming_that_Q_is_false_and_showing_that_P_is_false_under_that_assumption._3._Proof_by_Contradiction%3A_To_prove_a_statement_P_sometime_it_is_useful_to_assume_P_is_false_and_then_show_that_that_assumption_leads_to_a_contradiction%2C_i.e.%2C_that_you_can_prove_both_some_statement_and_its_negation._4._Proof_by_Cases%3A_If_you_know_that_either_P_or_Q_is_true%2C_and_want_to_show_R%2C_you_can_prove_R_by_considering_separate_cases._In_the_first_case%2C_assume_P_and_prove_R._In_the_second_case_assume_Q_and_show_R._Since_one_of_P_or_Q_is_known_to_be_true%2C_R_must_be_true_as_well._This_method_also_generalizes_to_situations_where_you_have_more_than_two_cases_to_consider._5._Proof_of_Univerality%3A_To_prove_that_a_statement_is_true_about_every_element_in_some_set%2C_let_x_be_an_arbitrary%2C_unspecified_element_of_that_set_and_prove_that_the_statement_is_true_about_x._6._Proof_of_Existence%3A_To_prove_that_a_mathematical_object_with_certain_properties_exists%2C_either_make_an_example_of_such_an_object_(i.e.%2C_construct_one)%2C_or_show_that_if_it_didn%E2%80%99t_exist_there_would_be_a_contradiction_(i.e.%2C_use_proof_by_contradiction)._7._Proof_by_Induction%3A_Let_P_(n)_be_a_statement_about_an_unspecified_natural_number_n._To_prove_that_P_is_true_for_all_natural_number_values_of_n%2C_show_P_(0)_is_true%2C_then_let_k_be_a_natural_number_and_assuming_P_(k)_is_true%2C_show_that_P_(k_%2B_1)_is_also_true._P_R_O_B_L_E_M_The_plane_is_divided_into_regions_by_finitely_many_straight_lines._Show_that_it_is_always_possible_to_color_the_regions_with_two_colors_so_that_adjacent_regions_are_never_the_same_color_(like_a_checkerboard)._%C2%A9_2022_KEN_MONKS_PAGE_9_of_23_The_Art_of_Problem_Solving_4_Tactics_Zeitz_identifies_the_following_tactics%2C_which_apply_to_a_wide_range_of_problems._Symmetry_Look_for_harmony_and_beauty%2C_whenever_you_investigate_a_problem._If_you_can_do_something_that_makes_things_more_harmonious_or_more_beautiful%2C_even_if_you_have_no_idea_how_to_define_these_two_terms%2C_then_you_are_often_on_the_right_track._%E2%80%93_Zeitz_Tactic%3A_When_a_problem_has_symmetry%2C_try_to_use_it._Try_to_maintain_that_symmetry_while_solving_the_problem._If_a_problem_doesn%E2%80%99t_have_symmetry%2C_but_you_wish_it_did%2C_try_to_introduce_symmetry_into_the_situation_if_possible._A_mathematical_object_(shape%2C_expression%2C_system_of_equations%2C_etc.)_is_symmetric_with_respect_to_some_action_or_operation_if_it_is_unchanged_by_the_action_or_operation._The_actions_that_do_this_are_called_the_symmetries_of_the_object._P_R_O_B_L_E_M_(MATHCOUNTS_2002_Workout_8%2C_number_8)_The_points_A%2C_B_and_C_lie_in_a_plane_and_have_coordinates_(6%2C_5)%2C_(2%2C_1)_and_(0%2C_k)%2C_respectively._What_value_of_k_makes_the_sum_of_the_lengths_of_segments_AC_and_BC_the_least_possible_value%3F_Examples_of_Symmetry_Groups_A_group_is_a_set_together_with_an_associative_binary_operator_on_that_set_that_has_an_identity_element_and_inverses_for_every_element_in_the_set._The_set_of_symmetries_of_an_object_often_forms_a_group._The_set_%7B1%2C_2%2C_3%2C_._._._%2C_n%7D_is_unchanged_by_permuting_its_elements._The_set_of_all_permutations_of_%7B1%2C_2%2C_3%2C_._._._%2C_n%7D_is_called_the_symmetric_group_Sn._%C2%A9_2022_KEN_MONKS_PAGE_10_of_23_The_Art_of_Problem_Solving_Geometric_Symmetry_Plane_geometric_figures_which_are_unchanged_by_reflection_across_a_line%2C_or_rotation_through_a_certain_angle%2C_or_inversion_in_a_circle%2C_or_translation_by_a_fixed_vector%2C_etc._are_said_to_be_symmetric_with_respect_to_that_line%2C_rotation%2C_inversion%2C_translation%2C_etc._The_set_of_all_isometries_(bijections_from_the_plane_to_itself_that_preserve_distance)_which_map_a_given_figure_to_itself_is_called_the_symmetry_group_for_that_figure._Symmetric_Functions_and_Polynomials_A_function_f_is_called_a_symmetric_function_in_n_variables_if_f_(x1%2C_x2%2C_._._._%2C_xn)_%3D_f__x%CF%83(1)_%2C_x%CF%83(2)_%2C_._._._%2C_x%CF%83(n)_%01_for_permutation_%CF%83_of_%7B1%2C_2%2C_3%2C_._._._%2C_n%7D_and_any_values_of_x1%2C_x2%2C_._._._%2C_xn._If_f_(x1%2C_x2%2C_._._._%2C_xn)_is_a_polynomial_in_x1%2C_x2%2C_._._._%2C_xn_we_say_f_(x1%2C_x2%2C_._._._%2C_xn)_is_a_symmetric_polynomial._e.g._here_are_some_symmetric_polynomials_in_two_variables%3A_x_%2B_y_x_2_%2B_y_2_x_2_y_%2B_xy2_x_3_%2B_x_2_y_%2B_xy2_%2B_y_3_System_of_Equations_A_system_of_equations_in_n_variables_is_a_symmetric_with_respect_to_its_variables_if_permuting_the_variables_leaves_the_system_unchanged._For_example%2C_x_%2B_y_%3D_z_x_%2B_z_%3D_y_y_%2B_z_%3D_x_In_such_situations_it_is_usually_in_your_best_interest_to_try_to_maintain_the_symmetry_in_whatever_algebraic_operations_you_do%2C_i.e.%2C_do_the_same_thing_to_all_three_equations_at_once_rather_than_operating_on_one_equation_at_a_time._For_example%2C_find_x_%2B_y_%2B_z_if_x%2C_y%2C_and_z_satisfy_the_above_system_of_equations._Similar_situations_can_arise_with_inequalities._%C2%A9_2022_KEN_MONKS_PAGE_11_of_23_The_Art_of_Problem_Solving_P_R_O_B_L_E_M_(Friendly_Competition_P1989-3)_Show_that_the_area_of_the_shaded_triangle_formed_by_the_side-trisectors_of_a_triangle_(as_shown)_is_1%2F7_of_the_area_of_the_whole_triangle._The_Extreme_Principle_Tactic%3A_Whenever_possible_put_the_elements_of_your_problem_in_some_order._Focus_on_the_largest_and_smallest_(i.e.%2C_extreme)_elements_in_this_order_as_they_may_be_constrained_in_interesting_ways._A_poset_(or_partially_ordered_set)_is_a_pair_(A%2CR)_where_A_is_a_set_and_R_is_a_relation_on_A_that_is_reflexive%2C_transitive%2C_and_antisymmetric._1._(reflexive)_%E2%88%80a_%E2%88%88_A%2C_aRa_2._(transitive)_%E2%88%80a%2C_b%2C_c_%E2%88%88_A%2C_aRb_and_bRc_%E2%87%92_aRc_3._(antisymmetric)_%E2%88%80a%2C_b_%E2%88%88_A%2C_aRb_and_bRa_%E2%87%92_a_%3D_b_Note_that_a_poset_might_not_have_a_largest_or_smallest_element._A_poset_(A%2CR)_is_well-ordered_if_and_only_if_it_satisfies_these_two_conditions%3A_1._(totally_ordered)_%E2%88%80a%2C_b_%E2%88%88_A%2C_aRb_or_bRa_2._Every_nonempty_subset_of_A_has_a_least_element._The_most_common_orderings_and_partial_orderings_encountered_in_problems_include%3A_%E2%80%A2_%3C_and_%E2%89%A4_on_sets_of_real_numbers_%E2%80%A2_%E2%8A%86_and_%E2%8A%8A_on_sets_of_sets_%E2%80%A2_alphabetical_and_lexicographic_ordering_on_strings_and_tuples_%E2%80%A2_ordering_polynomials_by_degree_%E2%80%A2_ordering_complex_numbers_by_absolute_value_%C2%A9_2022_KEN_MONKS_PAGE_12_of_23_The_Art_of_Problem_Solving_Note_that_you_can_often_order_your_elements_in_more_than_one_way%2C_so_that_you_should_use_the_strategy_of_staying_loose_when_considering_orderings_as_one_particular_ordering_might_be_more_useful_than_another._One_ordering_principle_that_can_be_very_helpful_when_dealing_with_sets_of_natural_numbers_is_actually_an_axiom_of_the_natural_numbers%3A_Well_Ordering_Principle%3A_Every_set_of_natural_numbers_has_a_least_element._P_R_O_B_L_E_M_(Zeitz_problem_1.1.4)_I_invite_10_couples_to_a_party_at_my_house._I_ask_everyone_present%2C_including_my_wife%2C_how_many_people_they_shook_hands_with._It_turns_out_that_everyone_questioned_%E2%80%93_I_didn%E2%80%99t_question_myself%2C_of_course_%E2%80%93_shook_hands_with_a_different_number_of_people._If_we_assume_that_no_one_shook_hands_with_his_or_her_partner%2C_how_many_people_did_my_wife_shake_hands_with%3F_(I_did_not_ask_myself_any_questions.)_P_R_O_B_L_E_M_Given_that_7_distinct_positive_integers_add_up_to_100%2C_prove_that_some_three_of_them_add_up_to_at_least_50._P_R_O_B_L_E_M_Given_2n%2B2_points_in_the_plane%2C_no_three_collinear%2C_prove_that_two_of_them_determine_a_line_that_separates_n_of_the_points_from_the_other_n._The_Pigeonhole_Principle_Tactic%3A_When_you_need_to_prove_something_exists_but_don%E2%80%99t_care_what_it_is%2C_consider_using_the_Pigeonhole_Principle_Pigeonhole_Principle%3A_If_you_have_n_pigeons_in_k_holes_some_hole_contains_at_least_l_n_k_m_pigeons_and_some_hole_contains_at_most_j_n_k_k_pigeons._%C2%A9_2022_KEN_MONKS_PAGE_13_of_23_The_Art_of_Problem_Solving_This_is_more_powerful_than_it_looks._Note_that_the_pigeonhole_principle_does_not_tell_you_which_hole_contains_the_number_of_pigeons_indicated%2C_nor_that_any_hole_contains_exactly_that_number_of_pigeons._It_is_just_tells_you_that_some_hole_satisfies_the_condition_given._P_R_O_B_L_E_M_Given_n_%2B_1_positive_integers_prove_that_there_are_two_of_them_whose_difference_is_divisible_by_n._P_R_O_B_L_E_M_Show_that_if_there_are_n_people_at_a_party%2C_then_two_of_them_know_the_same_number_of_people_(among_those_present%2C_assuming_%22knowing%22_is_symmetric)._P_R_O_B_L_E_M_Show_that_if_five_distinct_points_are_placed_inside_an_equilateral_triangle_of_side_length_two%2C_there_are_two_distinct_points_that_are_a_distance_less_than_or_equal_to_one_apart._P_R_O_B_L_E_M_Prove_that_there_exist_integers_a%2C_b%2C_c_not_all_zero_and_each_of_absolute_value_less_than_one_million_such_that_%0C_%0C_%0C_%0C_a_%2B_b_%E2%88%9A_2_%2B_c_%E2%88%9A_3_%0C_%0C_%0C_%0C_%3C_10%E2%88%9211_Invariants_and_Monovariants_Tactic%3A_When_there_are_too_many_things_to_keep_track_of%2C_look_for_an_invariant_or_monovariant._An_invariant_is_any_quantity_or_property_that_remains_the_same_under_some_operation_or_is_the_same_for_all_elements_of_a_set._Parity_and_congruence_are_common_invariants._%C2%A9_2022_KEN_MONKS_PAGE_14_of_23_The_Art_of_Problem_Solving_Example%3A_Consider_a_point_P_%3D__x%2C_y_%01_on_the_unit_circle._As_P_moves_around_the_circle_its_x_and_y_coordinates_change%2C_but_the_value_of_x_2_%2B_y_2_is_always_1._Example%3A_For_example%2C_the_operations_f(n)_%3D_3n_%2B_1_always_maps_an_integer_to_another_integer_having_the_same_parity_after_two_iterations._P_R_O_B_L_E_M_There_is_one_stone_at_each_vertex_of_a_square._We_are_allowed_to_change_the_number_of_stones_according_to_the_following_rule%3A_We_may_take_away_any_number_of_stones_from_any_vertex_and_add_twice_as_many_stones_to_the_pile_at_one_of_the_adjacent_vertices._Is_it_possible_to_get_2004%2C_2003%2C_2005%2C_and_2004_stones_at_consecutive_vertices_after_a_finite_number_of_moves%3F_P_R_O_B_L_E_M_Let_there_be_nine_lattice_points_in_a_three-dimensional_Euclidean_space._Show_there_is_a_lattice_point_on_the_interior_of_one_of_the_line_segments_joining_two_of_these_points._P_R_O_B_L_E_M_Is_it_possible_to_start_with_a_knight_in_some_corner_of_a_chessboard_and_reach_the_opposite_corner_by_a_sequence_of_legal_moves_that_pass_through_every_square_exactly_once%3F_A_monovariant_is_any_quantity_that_strictly_increases_or_strictly_decreases_under_some_operation._A_monovariant_often_has_positive_integer_values_and_is_strictly_decreasing%2C_so_that_after_finitely_many_steps_it_is_zero._Example%3A_In_the_stone-and-square_problem_above%2C_the_total_number_of_stones_on_all_four_corners_is_a_monovariant._It_goes_up_by_one_for_each_stone_removed._P_R_O_B_L_E_M_At_a_round_table_are_2004_girls%2C_playing_a_game_with_a_deck_of_n_cards._Initially_one_girl_holds_all_the_cards._At_each_turn%2C_if_at_least_one_girl_holds_at_least_two_of_these_cards%2C_one_of_these_girls_must_pass_a_card_to_each_of_her_two_neighbors._The_game_ends_when%2C_and_only_when_each_girl_is_holding_at_most_one_card._Prove_that_the_game_will_end_if_and_only_if_n_%3C_2004._%C2%A9_2022_KEN_MONKS_PAGE_15_of_23_The_Art_of_Problem_Solving_5_Crossover_Tactics_Graph_Theory_A_graph_is_a_is_a_pair_of_sets_(N%2C_E)_where_E_is_a_set_of_one_or_two_element_subsets_of_N._A_directed_graph_(or_digraph)_is_a_pair_of_sets_(N%2C_E)_where_E_%E2%8A%86_N_%C3%97_N._The_elements_of_N_are_called_nodes_and_the_elements_of_E_are_called_edges._Graphs_and_digraphs_are_usually_represented_pictorially_by_drawing_the_nodes_as_points_and_connecting_nodes_x_and_y_with_an_arrow_whenever_x%2C_y_%01_or_a_curve_if_%08_x%2C_y_%09_is_an_edge._The_positioning_of_the_nodes_and_the_shapes_of_the_edges_drawn_is_irrelevant._Graphs_vs._Relations%3A_Since_every_relation_on_N_can_be_represented_as_a_subset_of_N_%C3%97_N%2C_the_set_E_in_any_digraph_is_a_relation_on_N._Similarly%2C_given_a_relation_R_on_N_we_can_construct_the_digraph_of_that_relation._Thus_both_are_just_different_representations_of_the_same_mathematical_concept._Symmetric_relations_can_be_represented_by_a_graph_instead_of_a_digraph._This_explains_why_graph_theory_is_a_crossover_tactic!_Graph_mini-lexicon_%E2%80%A2_If_N_is_finite_we_say_the_graph_or_digraph_(N%2C_E)_is_also_finite._The_number_of_nodes_in_a_finite_graph_or_digraph_is_called_the_order_of_the_graph._%E2%80%A2_A_graph_or_digraph_is_simple_if_it_has_no_edges_from_a_point_to_itself._%E2%80%A2_(N%E2%80%B2_%2C_E_%E2%80%B2_)_is_a_subgraph_of_graph_(N%2C_E)_if_(N%E2%80%B2_%2C_E_%E2%80%B2_)_is_a_graph_and_N%E2%80%B2_%E2%8A%86_N_and_E_%E2%80%B2_%E2%8A%86_E._%E2%80%A2_A_path_from_x1_to_xn_in_a_graph_is_a_sequence_of_nodes_x1%2C_x2%2C_._._._%2C_xn_having_an_edge_between_any_two_consecutive_terms_in_the_sequence_(these_edges_are_said_to_be_in_the_path)_where_no_edge_is_in_the_path_more_than_once._In_the_case_where_x1_%3D_xn_we_say_the_path_is_a_cycle._%E2%80%A2_A_graph_having_a_path_between_every_two_vertices_is_a_connected_graph._%E2%80%A2_A_connected_graph_with_no_cycles_is_called_a_tree._%E2%80%A2_The_degree_of_a_node_in_a_finite_graph_is_the_number_of_edges_connected_to_that_node._In_a_finite_digraph_we_have_the_out-degree_and_in-degree_of_each_node_which_is_the_number_of_edges_leaving_and_entering_the_node%2C_respectively._%E2%80%A2_A_bipartite_graph_is_a_graph_whose_nodes_can_be_partitioned_into_two_subsets_U_and_V_such_that_every_edge_connects_an_element_of_U_to_an_element_of_V._%C2%A9_2022_KEN_MONKS_PAGE_16_of_23_The_Art_of_Problem_Solving_%E2%80%A2_An_Eulerian_path_is_a_path_in_a_graph_that_contains_every_edge._%E2%80%A2_A_Hamiltonian_path_is_a_path_in_a_graph_that_contains_every_vertex._%E2%80%A2_A_graph_is_complete_if_there_is_an_edge_connecting_every_pair_of_nodes._A_bipartite_graph_is_complete_if_there_is_an_edge_between_every_pair_(u%2C_v)_with_u_%E2%88%88_U_and_v_%E2%88%88_V._Playbook_Facts_about_Graphs_Connected_components%3A_The_relation_%22there_is_a_path_from_x_to_y%22_partitions_a_graph_into_a_disjoint_union_of_connected_subgraphs._These_are_called_the_connected_components_of_the_graph._Handshake_Lemma%3A_In_any_finite_graph_the_sum_of_the_degrees_of_the_nodes_is_twice_the_number_of_edges._Existence_of_Eulerian_paths_and_cycles%3A_A_graph_has_an_Eulerian_path_if_and_only_if_it_is_connected_and_the_number_of_nodes_of_odd_degree_is_either_two_or_zero._A_path_has_an_Eulerian_cycle_if_and_only_if_it_is_connected_and_every_vertex_has_even_degree._Existence_of_Hamiltonian_cycles_-_Dirac%E2%80%99s_Theorem%3A_A_simple_graph_with_n_nodes_has_an_Hamiltonian_cycle_if_the_degree_of_every_node_is_at_least_n%2F2._Existence_of_Hamiltonian_cycles_-_Ore%E2%80%99s_Theorem%3A_A_simple_graph_with_n_nodes_has_an_Hamiltonian_cycle_if_whenever_two_nodes_are_not_connected_by_an_edge_the_sum_of_their_degrees_is_at_least_n._P_R_O_B_L_E_M_There_are_59_people_at_a_party._Prove_that_someone_shook_hands_an_even_number_of_times._P_R_O_B_L_E_M_Show_that_in_any_group_of_six_people_there_are_either_three_who_are_mutual_friends_or_three_who_are_mutual_strangers._%C2%A9_2022_KEN_MONKS_PAGE_17_of_23_The_Art_of_Problem_Solving_P_R_O_B_L_E_M_There_are_n_people_at_a_party._For_any_two_people_at_the_party_who_are_not_friends%2C_the_sum_of_the_number_of_people_at_the_party_that_each_is_friends_with_is_at_least_n._Prove_everyone_at_that_party_can_be_seated_at_a_round_table_so_that_nobody_sits_next_to_anyone_who_is_not_their_friend._(You_may_assume_that_nobody_is_their_own_friend.)_Note%3A_this_is_actually_the_proof_of_Ore%E2%80%99s_Theorem_in_disguise!_Complex_Numbers_Let_C_%3D_R2_._For_each_(x%2C_y)_%E2%88%88_C_we_formally_write_x%2C_y_%01_%3D_x_%2B_yi._This_form%2C_x_%2B_yi%2C_is_called_the_standard_form_of_the_complex_number_x%2C_y_%01_._Let_x_%2B_yi%2C_a_%2B_bi_%E2%88%88_C._1._x_%2B_yi_%3D_x_%E2%88%92_yi._(This_is_called_the_complex_conjugate.)_2._%0C_%0C_%0Cx_%2B_yi_%0C_%0C_%0C_%3D_p_x_2_%2B_y_2_._(This_is_called_the_complex_norm.)_3._Arg(x_%2B_yi)_%3D_the_angle_in_%5B0_._._._2%CF%80)_of_(x%2C_y)_in_polar_form_(not_defined_for_x_%3D_y_%3D_0)._(This_is_called_the_Argument_of_x_%2B_yi.)_4._Re(x_%2B_yi)_%3D_x._(This_is_called_the_real_part_of_x_%2B_yi.)_5._Im(x_%2B_yi)_%3D_y._(This_is_called_the_imaginary_part_of_x_%2B_yi.)_6._(x_%2B_yi)_%2B_(a_%2B_bi)_%3D_(x_%2B_a)_%2B_(y_%2B_b)i._(This_is_the_definition_of_addition_in_C.)_7._(x_%2B_yi)(a_%2B_bi)_%3D_(xa_%E2%88%92_yb)_%2B_(ya_%2B_xb)i.(This_is_the_definition_of_multiplication_in_C.)_Notation._We_can_abbreviate_0_%2B_yi_as_yi%2C_x_%2B_0i_as_x%2C_x_%2B_1i_as_x_%2B_i%2C_and_x_%E2%88%92_1i_as_x_%E2%88%92_i_with_no_ambiguity_in_the_above_definitions._With_this_notation_i_%3D_(0%2C_1)_and_i_2_%3D_%E2%88%921._It_is_easy_to_verify_that_the_usual_laws_of_addition_and_multiplication_(associative%2C_commutative%2C_distributive%2C_identity%2C_etc.)_hold_for_the_complex_numbers_as_well._Let_%CE%B8_%E2%88%88_R._Then_e_i%CE%B8_%3D_cos_%CE%B8_%2B_i_sin_%CE%B8_Let_x_%2B_yi_%E2%88%88_C_%E2%88%92_%7B0%7D._The_standard_polar_form_of_x_%2B_yi_is_rei%CE%B8_where_r_%3D_%0C_%0C_%0Cx_%2B_yi_%0C_%0C_%0C_and_%CE%B8_%3D_Arg(x_%2B_yi)._The_distance_between_two_complex_numbers_z%2C_w_is_denoted_d(z%2C_w)_and_is_defined_to_be_d(z%2C_w)_%3D_%7Cz_%E2%88%92_w%7C._Theorem_1._Let_%CE%B8%2C_%CE%B3_%E2%88%88_R._1._e_i%CE%B8_e_i%CE%B3_%3D_e_i(%CE%B8%2B%CE%B3)_._2._%0C_%0C_%0Ce_i%CE%B8_%0C_%0C_%0C_%3D_1._%C2%A9_2022_KEN_MONKS_PAGE_18_of_23_The_Art_of_Problem_Solving_3._e_i%CE%B8_%3D_e_i(%E2%88%92%CE%B8)_._Theorem_2._Let_z%2C_z1%2C_z2_%E2%88%88_C._1._%7Cz1z2%7C_%3D_%7Cz1%7C_%7Cz2%7C_2._z1z2_%3D_z1_z2%2C_i.e.%2C_the_conjugate_of_a_product_is_the_product_of_conjugates._3._z1_%2B_z2_%3D_z1_%2B_z2%2C_i.e.%2C_the_conjugate_of_a_sum_is_the_sum_of_the_conjugates._4._z_z_%3D_%7Cz%7C_2_5._%7Cz%7C_%3D_%0C_%0C_%0Cz_%0C_%0C_%0C_6._If_z_%3D_rei%CE%B8_in_polar_form%2C_then_z_%3D_rei(%E2%88%92%CE%B8)_A_transformation_of_a_set_S_is_a_bijection_from_S_to_S._Remark._In_other_branches_of_mathematics_a_transformation_of_S_is_often_called_a_permutation_of_S._Useful_Geometric_Transformations_Let_w_%E2%88%88_C_and_%CE%B8%2C_k_%E2%88%88_R._Transformation_Description_T(z)_%3D_z_%2B_w_Translation_by_w_T(z)_%3D_e_i%CE%B8_z_Rotation_by_%CE%B8_radians_counterclockwise_about_the_origin_T_(z)_%3D_z_Reflection_across_the_x-axis_T(z)_%3D_kz_Homothety_by_positive_factor_k_with_respect_to_the_origin_T(z)_%3D_1_z_Inversion1_with_respect_to_the_unit_circle_Remark._You_can_compose_these_functions_to_obtain_many_useful_transformations!_P_R_O_B_L_E_M_(Arithmetic)_Prove_that_if_an_integer_can_be_written_as_a_sum_of_two_squares%2C_then_so_can_any_positive_integer_power_of_that_integer._P_R_O_B_L_E_M_(Algebra)_Factor_z_5_%2B_z_%2B_1_(as_a_polynomial_with_integer_coefficients)._1_Inversion_is_a_transformation_of_the_extended_complex_plane_C_%2B_%3D_C%E2%88%AA%7B%E2%88%9E%7D_with_1_0_%3D_%E2%88%9E_and_1_%E2%88%9E_%3D_0._%C2%A9_2022_KEN_MONKS_PAGE_19_of_23_The_Art_of_Problem_Solving_P_R_O_B_L_E_M_(Trigonometry)_Express_cos(5%CE%B8)_in_terms_of_cos(%CE%B8)._P_R_O_B_L_E_M_(Geometry)_Let_ABCD_be_a_convex_quadrilateral_and_construct_a_square_on_each_side_lying_outside_of_the_quadrilateral._Show_that_the_two_line_segments_connecting_the_centers_of_the_two_pairs_of_opposite_squares_are_the_same_length_and_perpendicular_to_each_other._Generating_Functions_A_generating_function_is_a_clothesline_on_which_we_hang_up_a_sequence_of_numbers_for_display._%E2%80%93_Herbert_Wilf_(Generatingfunctionology)_The_generating_function_of_a_sequence_of_integers_a0%2C_a1%2C_a2%2C_._._._is_the_formal_power_series_a0_%2B_a1x_%2B_a2x_2_%2B_%C2%B7_%C2%B7_%C2%B7_%2B_anx_n_%2B_%C2%B7_%C2%B7_%C2%B7_Remark._A_generating_function_defines_a_function_f(x)_%3D_a0_%2B_a1x_%2B_a2x_2_%2B_%C2%B7_%C2%B7_%C2%B7_%2B_anx_n_%2B_%C2%B7_%C2%B7_%C2%B7_defined_for_all_x_for_which_the_series_converges._Remark._Variations_such_as_X%E2%88%9E_n%3D0_an_n!_x_n_are_sometimes_useful._Geometric_Series%3A_1_%2B_x_%2B_x_2_%2B_x_3_%2B_%C2%B7_%C2%B7_%C2%B7_%3D_1_1%E2%88%92x_Manipulating_ordinary_generating_functions_Let_A(x)_%3D_X%E2%88%9E_n%3D0_anx_n_and_B(x)_%3D_X%E2%88%9E_n%3D0_bnx_n_1._A(x)_%3D_B(x)_if_and_only_if_an_%3D_bn_for_all_n._%C2%A9_2022_KEN_MONKS_PAGE_20_of_23_The_Art_of_Problem_Solving_2._A(x)_(1%E2%88%92x)_%3D_P%E2%88%9E_n%3D0_Pn_k%3D0_ak_%01_x_n_3._A(x)B(x)_%3D_P%E2%88%9E_n%3D0_Pn_k%3D0_akbn%E2%88%92k_%01_x_n_4._xA%E2%80%B2_(x)_%3D_P%E2%88%9E_n%3D0_nanx_n_5._R_A(x)_dx_%3D_C_%2B_P%E2%88%9E_n%3D1_an%E2%88%921_n_x_n_Partial_Fraction_Decomposition%3A_If_p(x)_%E2%88%88_R_%5Bx%5D_has_degree_less_than_k%2B2mandl1(x)_%C2%B7_%C2%B7_%C2%B7_lk(x)_%E2%88%88_R_%5Bx%5D_are_irreducible_linear_polynomials_and_q1(x)_%C2%B7_%C2%B7_%C2%B7_qm_(x)_%E2%88%88_R_%5Bx%5D_are_irreducible_quadratic_polynomials_then_there_exist_real_numbers_A1%2C_._._._%2C_Ak_%2C_B1%2C_._._._%2C_Bk_%2CC1%2C_._._._%2C_Ck_such_that_p(x)_l1(x)_%C2%B7_%C2%B7_%C2%B7_lk(x)q1(x)_%C2%B7_%C2%B7_%C2%B7_qm(x)_%3D_A1_l1_(x)_%2B_%C2%B7_%C2%B7_%C2%B7_%2B_Ak_lk(x)_%2B_B1x_%2B_C1_q1(x)_%2B_%C2%B7_%C2%B7_%C2%B7_%2B_Bmx_%2B_Cm_qm(x)_Taylor_Series%3A_f(x)_%3D_P%E2%88%9E_n%3D0_f_(n)_(0)_n!_x_n_P_R_O_B_L_E_M_Let_a0%2C_a1%2C_a2%2C_._._._be_a_sequence_of_positive_integers_satisfying_a0_%3D_1_and_an_%3D_2an%E2%88%921_%2B_3_n_for_n_%E2%89%A5_1._Find_a_closed_formula_for_an._P_R_O_B_L_E_M_Prove_the_hockey_stick_identity_using_generatingfunctionological_methods.__k_k_!_%2B__k_%2B_1_k_!_%2B__k_%2B_2_k_!_%2B_%C2%B7_%C2%B7_%C2%B7_%2B__n_k_!_%3D__n_%2B_1_k_%2B_1_!_P_R_O_B_L_E_M_Let_Cn_be_the_number_of_ways_to_distribute_n_cookies_to_three_children_so_that_no_child_has_fewer_than_two_cookies_or_more_than_four_cookies%2C_and_there_is_no_shortage_of_cookies_or_cookies_left_over_after_the_distribution._Compute_Cn_for_all_n._5.1_Tools_Thinking_on_your_feet%3A_Mental_Arithmetic_Problem_solvers_become_adroit_at_mental_arithmetic._It_is_something_that_is_developed_over_time_as_you_practice._Along_the_way_you_will_learn_and_develop_many_tricks_for_%C2%A9_2022_KEN_MONKS_PAGE_21_of_23_The_Art_of_Problem_Solving_doing_arithmetic_in_your_head._Here_is_a_list_of_a_few_common_ones._There_are_plenty_more!_We_will_go_over_them_in_class._Prime_Factorization_is_your_best_friend!_The_problem_solver_often_prefers_to_think_of_a_positive_integer_as_a_product_of_primes%2C_not_as_its_base_ten_representation._%E2%80%A2_Fundamental_Theorem_of_Arithmetic%3A_Every_positive_integer_n_can_be_written_uniquely_as_a_product_of_prime_powers_in_increasing_order_of_the_primes%2C_i.e.%2C_there_is_a_unique_sequence_of_nonnegative_integer_exponents_e1%2Ce2%2C_._._._such_that_n_%3D_2_e1_3_e2_5_e3_7_e4_%C2%B7_%C2%B7_%C2%B7_%E2%80%A2_Divisibility_Tests%3A_are_very_useful_for_finding_the_prime_factorization_of_small_numbers_%E2%80%A2_Modular_Arithmetic%3A_divisibility_tests_also_give_the_remainder_for_integer_division._The_sum_of_the_remainders_is_the_remainder_of_the_sum_and_the_product_of_remainders_is_the_remainder_of_the_product._%E2%80%A2_Primality_Criteria%3A_a_positive_integer_p_is_prime_if_it_is_not_divisible_by_any_prime_n_such_that_n_2_%E2%89%A4_p._%E2%80%A2_Applications_to_Arithmetic%3A_multiplying_fractions%2C_reducing_fractions%2C_computing_gcd_and_lcm%2C_modular_arithmetic._It_is_often_useful_to_leave_numbers_in_their_prime_factorization_form_when_doing_arithmetic_rather_than_multiplying_the_prime_factorization_out_to_get_the_base_ten_representation._%E2%80%A2_Cancel%2C_cancel%2C_cancel%3A_when_multiplying_fractions_always_cancel_first!_%E2%80%A2_Primes_are_Good_Luck!%3A_always_check_if_your_phone_number%2C_home_address%2C_runner_number%2C_lotto_number%2C_etc._is_prime._They_are_good_luck!_Memorization%3A_not_fun%2C_but_useful_There_is_no_doubt_that_memorization_is_required_to_be_good_at_mental_arithmetic._Everyone_memorizes_their_times_tables_for_example._There_are_some_other_things_that_come_up_a_lot_and_can_be_quite_valuable_to_a_problem_solver._Of_course%2C_the_list_is_not_limited_to_these_items%2C_and_anything_you_do_beyond_this_list_is_certainly_going_to_be_of_value_to_you._But_these_are_some_of_the_common_things_that_problem_solvers_memorize._%E2%80%A2_Squares_%E2%80%A2_Cubes_%E2%80%A2_Factorials_%E2%80%A2_Powers_of_two_%E2%80%A2_Decimal_representations_of_fractions_%C2%A9_2022_KEN_MONKS_PAGE_22_of_23_The_Art_of_Problem_Solving_%E2%80%A2_Prime_numbers_%E2%80%A2_Useful_Prime_Factorizations_%E2%80%A2_Decimal_approximations_of_%CF%80%2Ce%2C_%E2%88%9A_2%2C_%E2%88%9A_3%2C_%E2%88%9A_5%2C_._._._%2C_ln(2)%2C_ln(3)%2C_._._._etc._%E2%80%A2_Pythagorean_triples_Specific_tricks_There_are_also_a_host_of_specific_tricks_that_only_apply_to_a_limited_situation%2C_but_that_situation_comes_up_often_enough_to_make_the_tricks_worthwhile._%E2%80%A2_Decimal_Representations_of_Sevenths_%E2%80%A2_Powers_of_eleven_%E2%80%A2_Squaring_two_digit_numbers_that_end_in_five_%E2%80%A2_Squaring_numbers_that_are_close_to_a_number_whose_square_you_know_%E2%80%A2_Multiplying_by_a_number_that_ends_in_9_or_1_%E2%80%A2_Left_to_right_operations%2C_especially_subtraction_%E2%80%A2_Fractions_are_usually_easier_than_decimals_%E2%80%A2_Famous_numbers_like_105_and_1001_%E2%80%A2_Converting_eventually_repeating_decimals_to_fractions_No%2C_seriously%2C_read_the_problem!_%E2%80%93_Leong_%C2%A9_2022_KEN_MONKS_PAGE_23_of_23 2026-07-31T18:08:28.591Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Biochemistry_is_the_study_of_the_chemical_processes_that_occur_within_living_organisms_%E2%80%94_essentially%2C_it's_where_chemistry_and_biology_meet_at_the_molecular_level._It_focuses_on_the_structure_and_function_of_the_four_major_classes_of_biomolecules%3A_proteins%2C_nucleic_acids%2C_carbohydrates%2C_and_lipids%2C_along_with_the_smaller_molecules_and_ions_that_support_them._Central_concerns_include_how_enzymes_catalyse_reactions_at_rates_that_would_otherwise_be_impossible_at_body_temperature%2C_how_cells_extract_and_store_energy_through_pathways_like_glycolysis_and_oxidative_phosphorylation%2C_and_how_genetic_information_flows_from_DNA_to_RNA_to_protein._The_field_underpins_much_of_modern_medicine_and_biotechnology%3A_understanding_a_metabolic_pathway_or_a_receptor's_binding_site_is_often_the_first_step_towards_designing_a_drug_that_targets_it._Techniques_such_as_X-ray_crystallography%2C_mass_spectrometry%2C_and_chromatography_let_biochemists_see_molecular_structures_and_track_reactions_in_detail%2C_turning_what_were_once_abstract_questions_about_%22the_chemistry_of_life%22_into_measurable%2C_testable_science. 2026-07-31T18:06:09.979Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Biochemistry_is_the_study_of_the_chemical_processes_that_occur_within_living_organisms_%E2%80%94_essentially,_it's_where_chemistry_and_biology_meet_at_the_molecular_level._It_focuses_on_the_structure_and_function_of_the_four_major_classes_of_biomolecules:_proteins,_nucleic_acids,_carbohydrates,_and_lipids,_along_with_the_smaller_molecules_and_ions_that_support_them._Central_concerns_include_how_enzymes_catalyse_reactions_at_rates_that_would_otherwise_be_impossible_at_body_temperature,_how_cells_extract_and_store_energy_through_pathways_like_glycolysis_and_oxidative_phosphorylation,_and_how_genetic_information_flows_from_DNA_to_RNA_to_protein._The_field_underpins_much_of_modern_medicine_and_biotechnology:_understanding_a_metabolic_pathway_or_a_receptor's_binding_site_is_often_the_first_step_towards_designing_a_drug_that_targets_it._Techniques_such_as_X-ray_crystallography,_mass_spectrometry,_and_chromatography_let_biochemists_see_molecular_structures_and_track_reactions_in_detail,_turning_what_were_once_abstract_questions_about_%22the_chemistry_of_life%22_into_measurable,_testable_science. 2026-07-31T18:06:02.860Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Protein 2026-07-31T18:02:53.687Z monthly 0.5 https://baiku.ai/https://thechessworld.com/articles/training-techniques/basic-chess-tips?srsltid=AfmBOor0DGmImpODs8YLoHX1lcGAICCbGd2PX-0Y7ro-mowcwqD1L20c 2026-07-31T17:58:34.837Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Serverside_confirmation 2026-07-31T16:32:27.116Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/NavIC 2026-07-31T16:14:29.823Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Stock_market 2026-07-31T15:02:29.892Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/History_of_Canada 2026-07-31T14:47:38.387Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Poolish 2026-07-31T14:30:20.588Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/NBDE_Dental_exam 2026-07-31T14:06:19.743Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Narwhal 2026-07-31T13:51:01.054Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Canada 2026-07-31T13:50:53.381Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Calisthenics 2026-07-31T13:28:25.104Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Orion's_Belt 2026-07-31T13:24:43.395Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Clarkson's_Farm 2026-07-31T13:22:31.965Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Renaissance 2026-07-31T12:48:15.213Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Probability 2026-07-31T12:47:40.358Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Quantum_mechanics 2026-07-31T12:46:57.169Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Immune_system 2026-07-31T12:46:13.163Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Electricity 2026-07-31T12:45:18.229Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Climate_change 2026-07-31T12:44:15.512Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Compound_interest 2026-07-31T12:43:36.150Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Memory 2026-07-31T12:42:55.273Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Somnolence 2026-07-31T12:42:31.590Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Volcano 2026-07-31T12:41:55.509Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/DNA 2026-07-31T12:40:49.734Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/apple-app-site-association 2026-07-31T07:13:05.288Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Paganism 2026-07-31T03:26:33.721Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Black_hole 2026-07-31T02:00:52.443Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Artificial_intelligence 2026-07-31T01:59:54.418Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Test 2026-07-31T01:57:02.322Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Large_language_model 2026-07-30T23:23:33.589Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/oral_pathology_for_NBDHE 2026-07-30T19:31:12.846Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Photosynthesis 2026-07-30T19:02:59.962Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Liquid-glass_transition 2026-07-30T18:28:16.480Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Xenophyophorea 2026-07-30T18:27:36.346Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Bioluminescence 2026-07-30T18:24:12.290Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Polytheism 2026-07-29T10:03:29.409Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Maypole 2026-07-29T03:39:28.000Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Oura_Health 2026-07-29T03:37:51.109Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Treasure_Island 2026-07-28T18:53:57.885Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Pasteurization 2026-07-28T16:32:11.176Z monthly 0.5 https://baiku.ai/https://taxa.substack.com/p/how-to-survive-boiling-water 2026-07-28T14:55:34.459Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion 2026-07-28T13:45:20.000Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Democracy 2026-07-28T13:44:55.178Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Black_hole 2026-07-28T13:41:39.537Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Osmosis 2026-07-28T13:37:12.201Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Photosynthesis_reaction 2026-07-28T13:36:48.479Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Interest 2026-07-28T13:24:19.849Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Periodontology 2026-07-28T13:06:18.111Z monthly 0.5 https://baiku.ai/https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%83%E0%A4%B7%E0%A5%8D%E0%A4%A3_%E0%A4%B5%E0%A4%BF%E0%A4%B5%E0%A4%B0 2026-07-28T12:46:26.489Z monthly 0.5 https://baiku.ai/https://scottsumner.substack.com/p/the-beauty-of-tautologies 2026-07-28T11:42:16.323Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/nbdhe_board 2026-07-28T11:17:48.952Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Blorptastic_widget_theory_xyz 2026-07-28T11:14:46.864Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Z%C3%A9phyrine_Qwintz_theorem_of_nothing 2026-07-28T03:41:56.702Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/National_Dental_Examining_Board_of_Canada 2026-07-28T02:50:30.638Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Harry_potter_7 2026-07-28T02:46:49.826Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Photometry 2026-07-28T02:45:36.968Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Angle 2026-07-28T01:04:26.557Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Moon 2026-07-28T01:01:34.125Z monthly 0.5 https://baiku.ai/https://fr.wikipedia.org/wiki/Lune 2026-07-28T01:01:15.908Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Nitrogen_cycle 2026-07-27T19:57:44.466Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Water_cycle 2026-07-27T19:33:19.108Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Heart 2026-07-27T19:21:52.929Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Space_exploration 2026-07-27T19:07:31.170Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Biodiversity 2026-07-27T14:49:23.684Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Dwarf_planet 2026-07-27T14:38:02.011Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Oceanography 2026-07-27T14:22:20.879Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Ecosystem 2026-07-27T13:38:36.977Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Marine_biology 2026-07-27T13:30:38.096Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Planet 2026-07-27T12:37:06.866Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Ecological_footprint 2026-07-27T08:19:42.406Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Greenhouse_gas_emissions 2026-07-27T08:07:51.806Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Planetary_system 2026-07-27T07:54:32.800Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Epigenetics 2026-07-27T06:33:53.368Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Computational_linguistics 2026-07-27T06:32:05.693Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Speech_recognition 2026-07-27T06:15:29.625Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Benito_Mussolini 2026-07-27T05:43:20.570Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Commercial_bank 2026-07-27T04:48:41.415Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Economic_growth 2026-07-27T04:27:06.607Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Mathematical_and_theoretical_biology 2026-07-27T03:43:47.254Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Systems_biology 2026-07-27T02:46:27.741Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Computational_biology 2026-07-27T01:17:22.007Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Conditional_probability 2026-07-27T00:57:18.319Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Biochemistry 2026-07-27T00:45:51.851Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Cell_biology 2026-07-27T00:16:22.430Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Bioinformatics 2026-07-27T00:05:57.985Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/DNA_sequencing 2026-07-26T23:48:12.809Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Galaxy_formation_and_evolution 2026-07-26T23:43:03.017Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Heredity 2026-07-26T23:28:04.362Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Dark_energy 2026-07-26T23:27:41.775Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Weakly_interacting_massive_particle 2026-07-26T23:23:03.211Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Economy 2026-07-26T23:20:00.589Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Investment 2026-07-26T23:19:05.157Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/White_dwarf 2026-07-26T23:13:28.251Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Pulsar 2026-07-26T22:59:04.946Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Supernova 2026-07-26T22:52:54.032Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Financial_market 2026-07-26T22:43:31.366Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Company 2026-07-26T21:56:35.830Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Genomics 2026-07-26T21:21:29.303Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Pollution 2026-07-26T20:28:23.162Z monthly 0.5 https://baiku.ai/https://es.wikipedia.org/wiki/Fotos%C3%ADntesis 2026-07-26T16:19:40.868Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Spacetime 2026-07-26T14:39:30.036Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Asteroid 2026-07-26T14:38:12.005Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Central_bank 2026-07-26T14:10:59.917Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation 2026-07-26T14:05:45.974Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Big_Bang 2026-07-26T13:53:42.964Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Barter 2026-07-26T11:17:52.316Z monthly 0.5 https://baiku.ai/https://es.wikipedia.org/wiki/Planeta 2026-07-26T11:01:32.306Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Ozone_depletion 2026-07-26T10:43:34.527Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Weather 2026-07-26T10:34:23.602Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Deflation 2026-07-26T10:24:17.797Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Andromeda_Galaxy 2026-07-26T10:16:47.030Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Economics 2026-07-26T10:10:05.501Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Atmosphere_of_Earth 2026-07-26T09:39:12.587Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Dark_matter 2026-07-26T08:49:22.368Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Renewable_energy 2026-07-26T08:02:14.406Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Ecology 2026-07-26T07:36:48.063Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Inflation 2026-07-26T07:33:11.846Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Programming_language 2026-07-26T07:22:09.336Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Supply_and_demand 2026-07-26T07:02:57.371Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Milky_Way 2026-07-26T06:50:28.034Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Educational_technology 2026-07-26T06:18:59.248Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Cosmology 2026-07-26T06:06:15.616Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Neural_network 2026-07-26T05:30:38.816Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Gene_expression 2026-07-26T05:26:27.368Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/United_States_dollar 2026-07-26T04:56:24.376Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Natural_language_processing 2026-07-26T04:47:53.672Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Game_theory 2026-07-26T04:41:55.864Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Gravity 2026-07-26T04:40:23.013Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Sun 2026-07-26T04:11:25.480Z monthly 0.5 https://baiku.ai/https://es.wikipedia.org/wiki/Fusi%C3%B3n_nuclear 2026-07-26T03:56:11.465Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Robotics 2026-07-26T03:43:49.252Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Protein_biosynthesis 2026-07-26T03:38:39.845Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Ribosome 2026-07-26T03:07:15.394Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Star 2026-07-26T03:01:32.372Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Artificial_neural_network 2026-07-26T02:15:33.338Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Typography 2026-07-26T01:47:56.557Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Etymology 2026-07-26T01:43:44.768Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Latin 2026-07-26T01:42:55.376Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Computer_science 2026-07-26T00:17:17.598Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Machine_learning 2026-07-25T23:51:12.676Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/Ethics_of_artificial_intelligence 2026-07-25T23:23:56.817Z monthly 0.5 https://baiku.ai/https://en.wikipedia.org/wiki/RNA 2026-07-25T22:43:31.108Z monthly 0.5