MathLabs

Problem 3

An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. The following is a four-row anti-Pascal triangle containing every integer from 1 through 10: 42657183109\begin{array}{ccccccccccc} &&&&4&&&&\\ &&&2&&6&&&\\ &5&&7&&1&&&\\ 8&&3&&10&&9 \end{array} Does there exist an anti-Pascal triangle with 2018 rows which contains every integer from 1 to 1+2+⋯+20181+2+\cdots+2018 ?
Step 1 of 6: Set the size and the maximum
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n=2018,N=1+2+⋯+n=n(n+1)2n=2018,\qquad N=1+2+\cdots+n=\frac{n(n+1)}2
Detailed analysis

Assume for contradiction that the requested triangle exists. It has n rows and contains each of the N distinct positive integers from 1 through N exactly once.