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 5 of 6: Start the second lightning path
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m=⌊n2⌋−1=1008,rj≥n+j(1≤j≤m)m=\left\lfloor\frac n2\right\rfloor-1=1008,\qquad r_j\ge n+j\quad(1\le j\le m)
Detailed analysis

Let X and Y be the two bottom-row neighbors of B; one of them is the last sibling qn−1q_{n-1}, hence is already one of the numbers from 1 through n. Reflect the triangle if necessary so that B is on the right half of the bottom row. Complete the triangular region on the side of this small neighbor to an apex C, and follow arrows from C to their bottom-row endpoint D, exactly as in the standard lightning construction. The endpoint position of B guarantees at least ⌊n/2⌋−1\lfloor n/2\rfloor-1 arrows on this second path. Every sibling increment rjr_j on it is a positive entry distinct from A and the qiq_i, so none is among the numbers from 1 through n. Ordering these distinct increments gives the lower bounds rj≥n+jr_j\ge n+j.