Fun with Numbers #13 – Tetrahedral Numbers

Tetrahedral numbers (or triangular pyramidal numbers) are the number of balls it would take to stack a triangular pyramid n high. For example the fourth pyramidal number is 20 since it takes 20 balls to stack a triangular pyramid 4 high. Here are the first ten: 1, 4, 10, 20, 35, 56, 84, 120, 165 and 220.

A tetrahedron 5 balls high contains 35 balls in total which is the 5th tetrahedral number

Tetrahedral numbers can be generated by the formula n(n+1)(n+2)/6 or by adding up the first n triangular numbers (the number of balls that it takes to make a triangle). For example a pyramid 4 high is made up of four different triangles, each one a different height. The first triangle consists of 1 ball, the second one consists of 3 balls, the third of 6 and fourth of 20 thus 1+3+6+10 = 20. The numbers 1, 3, 6 and 10 are the first four triangular numbers.

In 1850 an English lawyer, politician and mathematician Sir Fredrick Pollock conjectures that all positive whole numbers are the sum of at most five Tetrahedral numbers. For example the number 34 = 10 + 20 + 4; 4, 10 and 20 are tetrahedral numbers. However Sir Pollock’s claim has yet to be proven and remains an open problem in number theory.

In 1878 it was proven that there are only 3 tetrahedral numbers that are perfect squares, the first, second and forty-eighth: 1 = 1^2, 4 = 2^2 and 19600 = 140^2.

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