Bogglers Solutions

By Scott Kim|Wednesday, May 01, 2002




Pythagorean Cutup

There are other solutions.





Triples

1. The next two such triples are {7, 24, 25} and {9, 40, 41}. The differences between consecutive square numbers step through all the odd numbers, which you can verify by counting the number of squares in each successive band of color in the diagram on page 82. So any odd number squared will be the difference between two square numbers. Stated mathematically, 2n + 1 represents any odd number and the first of such a Pythagorean triple. 2n2 + 2n is the second number, and 2n2 + 2n + 1 is the third. Because n can be any integer, an infinite number of solutions exist. {77, 2,964, 2,965} is such a triple.

2. The next two such triples are {10, 24, 26} and {12, 35, 37}. The differences between squares of numbers that differ by 2 step through all the multiples of 4. 2n represents any even number and the first in such a triple (n can't equal 1). n2 - 1 is the second number, and n2 + 1 the third. {100, 2,499, 2,501} is such a triple.

3. There are no Pythagorean triples in which 77 is the largest number. That leaves us with 77 as one of the two smaller numbers in a Pythagorean triple. If we call the two other numbers x and x + a (which differ by a), then 772 = (x + a)2 - x2 = 2ax + a2 = a(2x + a). Therefore, a must divide evenly into 772 and must be less than 77. The divisors of 772 less than 77 are 1, 7, 11, and 49, and the corresponding Pythagorean triples are {77, 2,964, 2,965}, {77, 420, 427}, {77, 264, 275}, and {77, 36, 85}.



Counterexamples

1. 52 + 2 = 33 (25 + 2 = 27)
2. 23 + 1 = 32 (8 + 1 = 9)
3. 32 + 24 = 52 (9 + 16 = 25)
4. 102 + 52 = 53 (100 + 25 = 125)
5. 63 + 83 = 93 - 1 (216 + 512 = 729 - 1)
6. 93 + 103 = 13 + 123 (729 + 1,000 = 1 + 1,728)





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