Sum of Three Squares

Find the four smallest distinct positive integers a,b,c,d such that a^2+b^2+c^2=d^2.
Source: NCTM Mathematics Teacher

Consider the square integers 1,4,9,16,25,\dots 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.
A=\left \{1,4,9\right \}
Number of three-element subsets
1+4+9=14  Not a solution

A=\left \{1,4,9,16\right \}
1+4+9=14  Repeat of previous step
1+4+16=21  No
1+9+16=26  No
4+9+16=29  No

A=\left \{1,4,9,16,25\right \}
4 subsets are repeat; 6 new ones are listed below
25+1+4=30 No
25+1+9=35  No
25+1+16=42  No
25+4+9=38  No
25+4+16=45  No
25+9+16=50  No

A=\left \{1,4,9,16,25,36\right \}
10 subsets are repeat; 10 are new
36+1+4=41  No
36+1+9=46  No
36+1+16=53  No
36+1+25=62  No
36+4+9=49  Yes
Though not necessary we list the remaining 5 subsets for completeness
36+4+16=56  No
36+4+25=65  No
36+9+16=61  No
36+9+25=70  No
36+16+25=77  No

Answer: 2,3,6,7


About mvtrinh

Retired high school math teacher.
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One Response to Sum of Three Squares

  1. Thomas Egense says:

    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.

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