**Exercise 1.** Prove Proposition 1.

**Exercise 2. **Prove Corollary 1.

**Exercise 3. **Here is a variation a la Sudoku on Al-Maghribî’s puzzle proposed by Andrew Simoson of King College:

The landowner labeled his trees 1 to 81 according to their fruitfulness. On a sandy region of his orchard, he drew a 9 × 9 grid. After much trial and error he succeeded in entering all 81 integers into the grid so that the column sums were all the same. Thus, the first son would receive the trees labeled as in the first column, the second son the trees labeled as in the second column, and so on. As he was admiring his solution and before he could write it on some valuable paper, an infrequent rain rendered some cell numbers illegible. He recovered the first row easily enough. But what about the empty cells in the grid below. Can you help him recover his solution?

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

15 | 16 | 17 | ||||||

26 | 22 | |||||||

28 | 30 | 31 | ||||||

45 | 43 | 38 | 41 | |||||

47 | 48 | 51 | 49 | |||||

60 | 56 | 57 | 58 | |||||

71 | 68 | 69 | 64 | |||||

75 | 80 | 73 | 78 |

Is the solution unique?

**Exercise 4.** Use Latin squares to find all genuinely different solutions to Ibn Hamza’s problem in cases of \(n=2\) and \(n=3.\) Can you conjecture as to the number of genuinely different solutions for \(n=4\)?

**Exercise 5.** Show that not every solution of Ibn Hamza’s problem is generated by a Latin square in the case \(n=4.\)

*Hint.*** **Show that the following solution can never be generated by a Latin square.

1 | 2 | 3 | 4 |

9 | 5 | 6 | 7 |

10 | 11 | 12 | 8 |

14 | 16 | 13 | 15 |