Conference rusure::math

42.0. "Absolute difference triangles" by HARE::STAN () Fri Feb 24 1984 13:11

In an absolute difference triangle, you start with n numbers
and then successively form new rows by taking the absolute value
of the difference between successive entries.

For example:

	6     14    15     3     13
           8      1    12     10
               7    11     2
                  4     9
                     5

This remarkable triangle was discovered by George Sicherman.
It has the property that it consists of consecutive integers.

It is conjectured that there is no absolute difference triangle
with consecutive integer entries with order larger than 5.
(The order of the triangle is the number of elements in the top row.)

Nelson has been shown (by computer) that no such triangle exists for
orders 6 and 7.

			References

Martin Gardner, Mathematical Games, Scientific American, 236(1977)129-136.

Charles W. Trigg, Absolute Difference Triangles, Journal of Recreational
	Mathematics 9(1976-1977)271-275.

Please clarify:  must the triangle contain the number 1, or does Nelson's
proof hold for any set of consecutive integers?  It would seem that in the
latter case one would have a much different and more complicated proof.

[corrected response]

Charles W. Trigg proves in the article mentioned that if the
elements of an absolute difference triangle are consecutive
integers beginning with b, then b=1.

The proof was not hard, but I don't feel like typing it in
now.  It's not immediately obvious.

Computer search shows no absolute difference triangles
with consecutive integers, for n=6..12.

A simple proof shows there is none for n=14, or n=30.
Hint: Consider the parity of the numbers.