A Multiplication Based Logic Puzzle

Even though 937 is a prime number, 937 tiny rectangles can be arranged into this beautiful star. Why?

937 is the 13th star number because 6(13)(13 – 1) + 1 = 937.

It is also the 13th star number because it is 12 times the 12th triangular number plus one: Look at this pattern:

The first star number is 12 times the 0th triangular number plus 1. Thus, 12(0) + 1 = 1 (1 yellow rectangle in the center)
The second star number is 12 times the 1st triangular number plus 1. Thus, 12(1) + 1 = 13 (12 green + 1 yellow rectangle in the center)
The third star number is 12 times the 2nd triangular number plus 1. Thus, 12(3) + 1 = 37 (24 blue + 12 green + 1 yellow rectangle in the center)
and so on. . .until
The thirteen star number is 12 times the 12th triangular number plus 1. Thus, 12(78) + 1 = 937 (144 yellow + 132 orange + 120 red + 108 purple + 96 blue + 84 green + 72 yellow + 60 orange + 48 red + 36 purple + 24 blue + 12 green + 1 yellow rectangle in the center)

I made the star so that it consists of one tiny rectangle in the center surrounded by 6 triangles with 78 (the 12th triangular number) rectangles each with another 6 triangles of the same size to form the 6 points of the star.

I very much enjoyed making this star. If you look closely you will see thirteen concentric stars in it following the pattern yellow, green, blue, purple, red, and orange repeated. I added star outlines to make the three smallest stars easier to see.

I think the graphic says a lot about the number 937 all by itself. I hope you enjoy looking at it.

Here’s a little more about the number 937:

24² + 19² = 937, so 937 is the hypotenuse of a Pythagorean triple:
215-912-937 which can be calculated from 24² – 19², 2(24)(19), 24² + 19²

937 is also a palindrome in three other bases:
1021201 in BASE 3 because 1(3⁶) + 0(3⁵) + 2(3⁴) + 1(3³) + 2(3²) + 0(3¹) + 1(3⁰) = 937
1F1 in BASE 24 (F is 15 in base 10) because 1(24²) + 15(24¹) + 1(24⁰) = 937
1A1 in BASE 26 (A is 10 in base 10) because 1(26²) + 10(26¹) + 1(26⁰) = 937

  • 937 is a prime number.
  • Prime factorization: 937 is prime.
  • The exponent of prime number 937 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 937 has exactly 2 factors.
  • Factors of 937: 1, 937
  • Factor pairs: 937 = 1 × 937
  • 937 has no square factors that allow its square root to be simplified. √937 ≈ 30.6104557

How do we know that 937 is a prime number? If 937 were not a prime number, then it would be divisible by at least one prime number less than or equal to √937 ≈ 30.6. Since 937 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 937 is a prime number.

Here’s another way we know that 937 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 24² + 19² = 937 with 24 and 19 having no common prime factors, 937 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √937 ≈ 30.6. Since 937 is not divisible by 5, 13, 17, or 29, we know that 937 is a prime number.

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