153 – a triangular number

A triangular number is one that can be represented by dots arranged in a triangle, where each row has one fewer dots than the one below it.  The layout can be as a right-angled triangle, or as an equilateral one as illustrated on the left.

Mathematically, the n^th  triangular number is the total of the first n natural numbers, so is 1 +2 +3 +….+n.  The formula is n(n+1)/2.  Adding together any two consecutive triangular numbers gives a square number.

153 is the 17th triangular number, so is 17 x 18 / 2.  It is unusual because if you reverse the digits to give 351 it’s also a triangular number: 26 x 27 / 2 = 351.  If you add 100 to 153 you again get a triangular number: 22 x 23 / 2 = 253.  Also, if you knock the last digit off 153 you get 15, which is also triangular, and knocking the last digit off that gives 1 – another triangular number.  The only other three-digit number with this property is 666, which of course is also Biblically significant.  In both cases there’s a near miss for being able to add another number on the end and get a triangular one – both 1540 and 6670 are triangular.

Fascinating fact
As explained above, adding together the first n natural numbers produces the n^th triangular number. If you instead add the cubes of the first n natural numbers you get the square of the n^th triangular number.  For example, 1³ +2³ +3³ + 4³ + 5³ =
1 + 8 + 27 + 64 +125 = 225 = 15² = (1 + 2 + 3 + 4 + 5)².