Given three adjacent squares,

a + b = c.

“Beauty is the first test,” wrote G.H. Hardy. “There is no permanent place in the world for ugly mathematics.”

The lowly 3×3 magic square has modest pretensions — each row, column, and diagonal produces the same sum.

But perhaps it’s magicker than we suppose:

618² + 753² + 294² = 816² + 357² + 492² (rows)
672² + 159² + 834² = 276² + 951² + 438² (columns)
654² + 132² + 879² = 456² + 231² + 978² (diagonals)
639² + 174² + 852² = 936² + 471² + 258² (counter-diagonals)
654² + 798² + 213² = 456² + 897² + 312² (diagonals)
693² + 714² + 258² = 396² + 417² + 852² (counter-diagonals)

(R. Holmes, “The Magic Magic Square,” The Mathematical Gazette, December 1970)

More: Any of the equations above will still hold if you remove the middle digit or any two corresponding digits in each of the six addends.

Yet more: (6 × 1 × 8) + (7 × 5 × 3) + (2 × 9 × 4) = (6 × 7 × 2) + (1 × 5 × 9) + (8 × 3 × 4)

I think everything above will work for any rotation or reflection of the square (that is, for any normal 3×3 magic square). I haven’t checked, though.

Is a legal chess game possible in which all the pawns promote and each player has nine queens?

Yes — Freidrich Burchard of Germany and Friedrich Hariuc of Romania reached nearly identical solutions in 1980:

1. e4 f5 2. e5 Nf6 3. exf6 e5 4. g4 e4 5. Ne2 e3 6. Ng3 e2 7. h4 f4 8. h5 fxg3 9. h6 g5 10. Rh4 gxh4 11. g5 g2 12. g6 Bg7 13. hxg7 g1=Q 14. f4 h3 15. f5 h2 16. b4 a5 17. b5 a4 18. b6 a3 19. Bb2 Ra7 20. bxa7 axb2 21. a4 b5 22. a5 b4 23. a6 b3 24. c4 h1=Q 25. c5 h5 26. c6 Bb7 27. cxb7 c5 28. d4 c4 29. d5 Nc6 30. dxc6 c3 31. c7 c2 32. c8=Q c1=Q 33. b8=Q Qc7 34. a8=Q d5 35. a7 d4 36. Nc3 dxc3 37. Qa6 c2 38. Qa8b7 c1=Q 39. a8=Q Qd5 40. gxh8=Q+ Kd7 41. g7 bxa1=Q 42. g8=Q b2 43. f7 b1=Q 44. f8=Q h4 45. f6 h3 46. f7 h2 47. Qfa3 h1=Q 48. f8=Q exf1=Q+

This may be the shortest possible such game.

Draw any quadrilateral and connect the midpoints of its sides.

You’ll always get a parallelogram.

Triangles, too, have perfection at heart.

The familiar Mercator projection is useful for navigation, but it exaggerates the size of regions at high latitudes. Greenland, for example, appears to be the same size as South America, when in fact it’s only one eighth as large.

An equal-area projection such as the Mollweide, below, distorts the shapes of regions but preserves their relative size. This reveals some surprising facts: Russia is larger than Antarctica, Mexico is larger than Alaska, and Africa is just mind-bogglingly huge — larger than the former Soviet Union, larger than China, India, Australia, and the United States put together.

Will you either answer no to this question or pay me a million dollars?

(Raymond Smullyan)

Doodling on a napkin in 1958, mathematician Norman L. Gilbreath noticed something odd. First he wrote down the first few prime numbers in a row. Then, on each succeeding row, he recorded the (unsigned) difference between each pair of numbers in the row above:

The first digit in each row (except the first) is 1. Will this always be true, no matter how many prime numbers we start with? It’s been borne out in computer searches extending to hundreds of billions of rows. But no one knows for sure.