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:

\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+\cdots+2018

Step 4 of 6: Locate the small numbers beside the path

Previous step Next step

\{A,q_1,\ldots,q_{n-1}\}=\{1,2,\ldots,n\},\qquad B=N

Detailed analysis

At each step, q_i is immediately beside the lightning path. In particular, at the last step one of the two bottom-row neighbors of B is one of the numbers from 1 through n. Every number from 1 through n has already been used as A or as such a neighboring increment, so no other increment on a suitably chosen path can be one of these small numbers.