# Solution to Triangles

#### by Linus Strese

This puzzle consists of a series of number pyramids with some squares being colored differently than the other squares. We begin by filling the white squares using standard number pyramid rules, where each square is the sum of the two squares below it.

 61 33 28 21 12 16 20 1 11 5
 213 52 161 146 94 67 84 62 32 35 41 43 19 13 22 5 16 25 18 1 4
 173 180 -7 20 19
 53 70 34 36 28 25 9 27 13 15 12 15
 477 432 79 19 20 5 14
 25 14 11 11 3 8 16 3 5
 60 24 2 19 5

After filling some of the white squares using the standard rules we notice that the purple squares have numbers that are smaller than the sum of the numbers below them. Using this we deduce that the purple squares represent subtracting the number below and to the right from the number below and to the left. (Note that one of the purple squares has a negative number, which invalidates the possibility that the purple squares are taking the absolute difference.)

 61 33 28 21 12 16 20 1 11 5
 213 52 161 146 94 67 84 62 32 35 41 43 19 13 22 5 16 25 18 1 13 9 4
 173 180 -7 20 19
 53 70 34 36 28 25 9 27 13 15 4 21 12 15
 477 432 79 19 20 5 14
 25 14 11 11 3 8 16 3 5
 60 24 2 19 5

Continuing to fill in the squares we notice that the numbers in the green squares are much bigger than the sum of the two numbers below them. Using this fact we deduce that the green squares represent multiplying the two numbers below. Assuming that the base must be filled with positive integers, we can now fill all of the squares.

 61 33 28 21 12 16 20 1 11 5
 162 213 −51 52 161 212 146 94 67 145 84 62 32 35 110 41 43 19 13 22 5 16 25 18 1 13 9 4
 4844 173 28 180 −7 35 20 9 16 19
 2503 1243 1260 53 1190 70 88 35 34 36 28 60 25 9 27 13 15 4 21 12 15
 3756 4365 609 3888 477 132 432 9 53 79 460 28 19 34 45 20 23 5 14 20 25
 5425 217 25 203 14 11 192 11 3 8 16 12 1 3 5
 104 60 44 3 20 24 2 1 19 5

Having filled all the pyramids we now read the bases using A1Z26 and get the cluephrase TAKE PYRAMID TIPS MODULO TWENTY PLACE BASE. We now take the number in the top square of each pyramid modulo twenty, insert them into the base of the final image and reapply the previously used rules.

 3 8 1 19 11 1 1 2 4 3 16 5 4