Can you make a star out of a hexagon? If you have 1261 tiny squares arranged as a centered hexagon, you can rearrange those 1261 tiny squares into a six-pointed star as illustrated below!

37 was the last centered hexagonal number that was also a star number.

Here are some more facts about the number 1261:

- 1261 is a composite number.
- Prime factorization: 1261 = 13 × 97
- The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1261 has exactly 4 factors.
- Factors of 1261: 1, 13, 97, 1261
- Factor pairs: 1261 = 1 × 1261 or 13 × 97
- 1261 has no square factors that allow its square root to be simplified. √1261 ≈ 35.51056

1261 is the sum of two squares two different ways:

30² + 19² = 1261

35² + 6² = 1261

1261 is the hypotenuse of FOUR Pythagorean triples:

420-1189-1261 calculated from 2(35)(6), 35² – 6², 35² + 6²

485-1164-1261 which is (5-12-**13**) times **97**

539-1140-1261 calculated from 30² – 19², 2(30)(19), 30² + 19²

845-936-1261 which is **13** times (65-72-**97**)

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I love this kind of puzzle.

I’m glad you liked it. You can’t make a star out of most hexagons, but this one is special indeed.