Some definitions:

• N is "triangular" if it is in the sequence 1, 3, 6, 10, 15, ...
• For a triangular number N, a "difference triangle" is a triangle of numbers from 1 to N with one number in the bottom row, two numbers in the next row, 3 numbers in the next row, and so on; each number must be the difference between the two numbers above it.
• Important: a different triangle must use all the numbers from 1 to N exactly once each!

For example, here are the two possible difference triangles for N=3:

3    2           2    3

   1                 1

Here is an example of a failed attempt at a difference triangle for N=6:

6    4    1

   2    3

      5

This almost works: 6-4=2, 4-1=3, but 3-2 is not 5.

(a) Can you find a difference triangle for N=6?

(b) Can you find more than one difference triangle for N=6?

(c) Can you convince me that you've found all the difference triangles for N=6?

(d) Can you do the above for N=10?

(e) Can you do the above for N=15?

You are welcome to send in your solution even if you haven't managed all of the above - part (e) in particular will be very difficult!