Solution to Triangles

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Answer: CHASKA

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

Reading the top row we get the answer CHASKA.