The little math puzzle of perfect squares that needs math logic support
What number added to 100 and 164 separately will make perfect squares each time? Support with math logic and deductive reasoning.
Time to solve 5 minutes.
You may get the answer in 10 seconds, but the challenge is in supporting the answer with math reasoning.
Solution to the puzzle of two perfect squares
Though I got the answer in 10 seconds from my sense of squares of numbers, I set my mind to the problem of finding supporting math logic.
First set of conclusions: Forming the roadmap
- Being a puzzle, the numbers 100 and 164 must have a hidden pattern in them. No puzzle works on random numbers.
- It takes a few seconds to identify the hidden pattern: 82 + 102 = 164. This should prove to be useful.
- The number to be added must be larger than (164 - 100) = 64. It follows, the second square for 164 must be more than 225 or 152.
Use the pattern hidden in the pair of numbers 100 and 64
Out of the three unknowns, the number to be added, the square of the number for 164 and for 100, if we know one, the other two will follow automatically.
Focus shifts to finding the number corresponding to the perfect square for 164, especially when it is already known to be greater than 15.
Assume Y as the number corresponding the to the square for 164 and Z for 100.
The unknown number to be added,
X =Y2 - 164 = Z2 - 100,
Or, Y2 - Z2 = (Y + Z)(Y - Z) = 64, where Y must be greater than 15.
The two product terms of 64 on the left side of the equation must each be a multiple of 2 or one of the two 1.
- The smaller term (Y - Z) cannot be 1: In that case Y and Z must be an odd-even integer pair and their addition will result in an odd number. This will violate the condition of multiple of 2.
- The product terms can either be 2, 32 or 4, 16.
With the constraint of the larger product term greater than 15, the smallest number that can fit in the shoes of the larger term is 17 (16 + 12 not equal to 32).
The pair of terms for the product result 64 can only be 2 and 32.
Final Solution to the pair of perfect squares puzzle
17 - 15 = 2 and 17 + 15 = 32,
And the number to be added,
152 - 100 = 125 = 172 - 164.
With hidden pattern identification and deductive reasoning in math, it is quick to get to the solution.
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