Weihnachtsmann rot

Challenge from 6. December

Game of Cubes

The weekend challenges can be solved without wasting points until Monday 11:59:59pm (CET). For more information see the rules of the game.

Enjoy the puzzle!

Authors: Jesper Nederlof, Frits Spieksma

Project: 4TU.AMI

Challenge:

Elf Kubo and Ruprecht play a game with N ≥ 4 wooden cubes. At the beginning of the game, all six faces of each of these N cubes are empty and unlabeled.

In the first phase of the game, the two players label the 6N faces of the cubes with integers from the range 1, 2, … , N. In every move, exactly one face of one cube is labeled. They take turns at moving with Ruprecht making the first move.

In the second phase of the game, they build a tower from the \( N \) wooden cubes. The first (and bottom-most) cube in the tower must carry the integer 1 on one of its faces, the second one the integer 2, the third cube the integer 3, and so on. Ruprecht and Kubo take turns at choosing a cube with Ruprecht picking the first (and hence bottom-most) cube in the tower. The game ends only if in the k-th move there is no cube with integer k available.

Ruprecht wins the game if at the end of the game the tower consists of all N cubes. Otherwise, Kubo is the winner. During both phases of the game, Kubo and Ruprecht always make the best possible moves.

For which values of N with 4 ≤ N ≤ 7 can Ruprecht enforce a win?

Artwork: Julia Nurit Schönnagel

PDF download

Possible answers:

  1. Ruprecht can enforce a win only for N = 4.

  2. Ruprecht can enforce a win only for N = 4, 5.

  3. Ruprecht can enforce a win only for N = 4, 6.

  4. Ruprecht can enforce a win only for N = 4, 7.

  5. Ruprecht can enforce a win only for N = 5, 6.

  6. Ruprecht can enforce a win only for N = 4, 5, 6.

  7. Ruprecht can enforce a win only for N = 4, 5, 7.

  8. Ruprecht can enforce a win only for N = 4, 6, 7.

  9. Ruprecht can enforce a win only for N = 5, 6, 7.

  10. Ruprecht can enforce a win for N = 4, 5, 6, 7.

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