[vc_row][vc_column][vc_column_text]In Pascal’s triangle the number 1 appears infnitely many times. The number 2 appears just once. The number 3 appears twice.
Can you prove that all numbers other than 1 will appear a fnite
amount of times?
What is the first number that appears exactly 3 times?
What is the first number that appears exactly 4 times?
Can you find a number that appears exactly 5 times?
Can you find a number that appears exactly 6 times?
Can you find a number that appears more than 6 times?[/vc_column_text][lvca_accordion][lvca_panel panel_title="Reveal Solution"]The last row that can contain n is the nth row (counting the initial 1 as the 0th row). n will have appeared a finite amount of times up to this point therefore all numbers other than 1 will appear a finite amount of times.
The first number to appear 3 times is 6, the first number to appear 4 times is 10.
Numbers that appear more than 4 times are much rarer. The smallest is 120 which appears 6 times. The smallest number that appears 8 times is 3003.
No examples have been found of numbers that appear exactly 5 times; however, it is not known whether any exist.
Singmaster’s conjecture states that there is a finite upper bound on the number of times a number can appear. It is thought that this upper bound is 10 or 12, though no such examples have been found therefore it could be as low as 8.[/lvca_panel][/lvca_accordion][vc_column_text]
Problem Page Coordinator: Claire Baldwin – Mathematics in Education and Industry
Acknowledgement: The IMA are indebted to MEI for sourcing and supplying Mathematics Today with these well-known puzzles.
First published in Mathematics Today (December 2017)
In Pascal’s triangle the number 1 appears infnitely many times. The number 2 appears just once. The number 3 appears twice.
Can you prove that all numbers other than 1 will appear a fnite
amount of times?
What is the first number that appears exactly 3 times?
What is the first number that appears exactly 4 times?
Can you find a number that appears exactly 5 times?
Can you find a number that appears exactly 6 times?
Can you find a number that appears more than 6 times?[/vc_column_text][lvca_accordion][lvca_panel panel_title="Reveal Solution"]The last row that can contain n is the nth row (counting the initial 1 as the 0th row). n will have appeared a finite amount of times up to this point therefore all numbers other than 1 will appear a finite amount of times.
The first number to appear 3 times is 6, the first number to appear 4 times is 10.
Numbers that appear more than 4 times are much rarer. The smallest is 120 which appears 6 times. The smallest number that appears 8 times is 3003.
No examples have been found of numbers that appear exactly 5 times; however, it is not known whether any exist.
Singmaster’s conjecture states that there is a finite upper bound on the number of times a number can appear. It is thought that this upper bound is 10 or 12, though no such examples have been found therefore it could be as low as 8.[/lvca_panel][/lvca_accordion][vc_column_text]
