Find the four smallest distinct positive integers such that .

Source: NCTM Mathematics Teacher

**SOLUTION**

Consider the square integers We form sets of square integers; find all possible three-element subsets of those sets; and see if the sum of the three elements is a square.

Number of three-element subsets

Not a solution

Repeat of previous step

No

No

No

subsets are repeat; new ones are listed below

No

No

No

No

No

No

subsets are repeat; are new

No

No

No

No

Yes

Though not necessary we list the remaining subsets for completeness

No

No

No

No

No

**Answer**:

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A similar (but much harder) puzzle is to find 4 square numbers, such that when adding any 3 of them, you also get a square number. A brute force computer search can instantly solve this, I am not sure it can be done by hand.

I wrote a blog post about a similar unsolved problem in recreational mathematics, which I did not solve.

http://www.thomas-egense.dk/math/Squares_on_a_cube.html