Numbers in a Triangle

Number Theory Level 3

In the triangle, each of the numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 \color{#D61F06} 1, 2, 3, 4, 5, 6, 7, 8 is placed into a different circle. The sums of the numbers on each of the three sides of the triangle are equal to the same number S S . The sum of all of the different possible values of S S is ?

88 76 90 85 67

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Samir Betmouni
Jul 28, 2018

This is as much as I can type in.

The numbers 1..8 add up to 36. Call the numbers in the corners a,b,c and their mean m. (Their sum is 3m)

Then 3s = 36 + 3m, because we have added the three corners twice.

So s = 12 + m

Also: m is between 2 and 7, inclusive (3m is between 6 and 21). In which case s is 14,15,16,17,18 or 19

But s=14 is impossible because m = 2 means (a,b,c) = (1,2,3) so the middle number on the base has to be atleast 9.

And s=18 is impossible, because that corresponds to m=6, requiring a+b+c = 18. This requires the middle number on the base to be the same as the number at the top.

15,16,17,19 are possible. (By construction)

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Cases 15 and 17 are also impossible

A Former Brilliant Member - 2 years, 5 months ago

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15 is possible. I've just constructed it. The left leg is 1482, the right leg is 1356. Leaving the base with 276. Note that the corners add up to 9, giving a mean of 3 as per the construction in my earlier answer.

Samir Betmouni - 2 years, 1 month ago

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