# Story where royalty on another planet uses a tetromino puzzle?

There's humans living on another planet, the royal family trains their children by giving them a puzzle where they have to fill a rectangle with tetrominos, and then with pentominos. It was a full novel, not a short story, but I don't think I read all of it so I don't remember much more.

• "Imperial Earth" by A.C. Clarke, maybe? The family was important but not royal, and the puzzle was only pentominoes (no tetraminos). Jul 1, 2023 at 21:46
• At the risk of sounding ignorant- what are tetraminos and pentaminos? Jul 2, 2023 at 2:05
• @ClaraDiazSanchez I'd say that was worth an answer. Duncan's family are the first family of Titan so while they aren't royalty they are a form of aristocracy. Jul 2, 2023 at 5:08
• @ClaraDiazSanchez The Last Theorem by Clarke mentions tetrominos and pentominos, so it could be a conflation of the two books. Jul 2, 2023 at 5:15
• @Nu'Daq - "a geometric shape composed of four squares, connected orthogonally" - Think Tetris. Jul 2, 2023 at 7:46

There are similarities between the points in the question, and the plot of Imperial Earth by Arthur C. Clarke. There are also differences... so the identification can only be tentative.

The novel deals with Duncan Makenzie, living on Titan, the moon of Saturn, which has been colonised. Although not really a member of a "Royal Family" as indicated in the question, he is a member of the so-called "First Family" - his grandfather, Malcolm, was the person who first settled Titan. The fortunes of Titan and the family are so tightly entwined that there is an unofficial motto "What’s good for the Makenzies is good for Titan."

Duncan is given an intelligence/psychological test by his grandmother: to fill rectangular shapes with the 12 pentomino pieces. He found solutions for the 6 x 10, 5 x 12 and 4 x 15 rectangles, but got stuck on the 3 x 20 case, and had to call for help from his friend Karl. One of the 6 x 10 solutions was pictured in the book's frontispiece. Throughout the rest of the book Duncan always takes a set of pentominos with him when he travels, even on his trip to Earth, which forms the main plot of the novel. Tetraminos, however, do not make any appearance.