The Magic Triangle Puzzle (Numbers 1-9) with 17 Inside | Minute Math

This is an interesting math puzzle. We have a triangle with 17 inside of it. There are 9 circles around the triangle. There are 4 circles on each side. Each circle can only have 1 number in it between 1-9. That number can only be used once. Each side adds to 17. Where do the numbers 1-9 go?

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Time Stamps:
00:00 - The Problem
00:46 - The Steps
02:17 - The Answer

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[mrblackwhite]

Is it possible with 20? I tried a lot of

[carstenmeyer]

More interesting final question -- if we do _not_ consider symmetric solutions equivalent, how many solutions are there?


*Answer:* There should be *3! * (2 * 2^3) = 96"* possible solutions.


*Proof:* Let *"ck"* be the number in corner *"k".* Summing up all three sides we get

*3 * 17 = (c1 + c2 + c3) + ∑_{k=1}^9 k = (c1 + c2 + c3) + 45*

*=> c1 + c2 + c3 = 3 * 17 - 45 = 6*

Considering *"ck"* are distinct digits, that is only possible if *"ck"* are some permutation of "1; 2; 3". The side with "1; 2" is missing *14* to get to *17.* There are two possible choices:

*14 = 5 + 9 => 12 = 4 + 8 => 13 = 6 + 7*
*14 = 6 + 8 => 12 = 5 + 7 => 13 = 4 + 9*

We have *3!* choices to arrange the corners, *2* possible solutions for each corner arrangement, and two choices to arrange the order of the middle digits for each of the three sides ∎

[owmyballs]

Hi and thanks for the video - how were you able to arrive at the number 17 for each side?