Number Pyramids

A number pyramid is created such that the number in the pink triangle is equal to the sum of pairwise products of the 3 neighboring numbers in yellow triangles. For instance, in the left pyramid above, 11 = 1 2 + 2 3 + 3 1 11 = 1\cdot 2 + 2\cdot 3 + 3\cdot 1 .

The right pyramid has number 4 4 on top and 1 , 3 , 5 , 7 , 9 1,3,5,7,9 in the remaining 5 yellow triangles.

If each of a , b , c a, b, c in the pink triangles is a composite number, compute a + b + c a+b+c .


The answer is 221.

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1 solution

X X
Jul 2, 2018

There are only 20 cases if we choose 3 numbers from 1,3,4,5,7,9.Only 7 cases produces composite numbers. ( 1 , 3 , 9 ) 39 (1,3,9)\rightarrow39 ( 1 , 4 , 7 ) 39 (1,4,7)\rightarrow39 ( 1 , 4 , 9 ) 49 (1,4,9)\rightarrow49 ( 3 , 4 , 9 ) 75 (3,4,9)\rightarrow75 ( 3 , 5 , 9 ) 87 (3,5,9)\rightarrow87 ( 3 , 7 , 9 ) 111 (3,7,9)\rightarrow111 ( 5 , 7 , 9 ) 143 (5,7,9)\rightarrow143 Notice the triple ( 1 , 7 , a ) (1,7,a) .Only a = 4 a=4 fits,so 1 1 and 7 7 is on the second row.Let 1 1 be on the left,and 7 7 on the right.

Notice the triple ( 5 , b , c ) (5,b,c) . 1 1 can't be adjacent to 5 5 ,so 5 5 must be on the bottom right corner.

Notice the bottom right triple ( 5 , 7 , d ) , d = 9 (5,7,d),d=9 ,so the bottom middle number is 9 9

So it should be like this 4 1 39 7 3 39 9 143 5 4\\1\quad{\tiny{39}}\quad7\\3\quad{\tiny{39}}\quad9\quad{\tiny{143}}\quad5

39 + 39 + 143 = 221 39+39+143=221

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