Let f(N,Q,A) denote the largest number of squares that are attacked exactly A times. Clearly f(N,Q,A) = 0 if Q<A or Q>N^{2}–A.

The 2×2 boards are boring, as each square attacks each other. Thus f(2,Q,A) = N^{2}–Q when Q=A, and 0 otherwise.

The boards containing one piece are also boring, as a queen on the edge of the board attacks the fewest squares, and a queen in the middle of the board attacks the most squares. Thus f(N,1,0) = N^{2}–3N+2 and f(N,1,1) = N^{2}–4N+4.

Computer evidence suggests the following formulas. Can you prove any of them? What other patterns can you find?

- f(N,2,0) = N
^{2}–5N+7 - f(N,2,1) = 7N–18 (for N≥6)
- f(N,2,2) = 12 (for N≥8)
- f(N,3,0) = N
^{2}–6N+9 (for N≥6) - f(N,3,3) = 5 (for N≥4)
- f(N,4,0) = 4 (N–3)/2 (N–2)/2 (for N≥5)
- f(N,4,2) = 6N–12 (for even N)
- f(N,4,3) = 14 (for N≥6)
- f(N,4,4) = 4 (for N≥4)
- f(N,5,1) = 9N–31 (for N≥4)
- f(N,5,4) = 8 (for N≥5)
- f(N,5,5) = 3 (for N≥5)
- f(N,6,5) = 4 (for N≥4)
- f(N,6,6) = 2 (for N≥4)

Here are the best known solutions. A yellow background means all vacant squares are attacked A times.

Q \ A | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

2 | 1 square 2 solutions | 5 squares 1 solution | 5 squares 1 solution | ||||||

3 | 3 squares 1 solution | 5 squares 1 solution | 4 squares 2 solutions | ||||||

4 | 1 square 3 solutions | 4 squares 4 solutions | 5 squares 2 solutions | 2 squares 3 solutions | |||||

5 | 3 squares 1 solution | 4 squares 4 solutions | 2 squares 3 solutions | 2 squares 1 solution | |||||

6 | 2 squares 2 solutions | 3 squares 2 solutions | 2 squares 4 solutions | 1 square 3 solutions | 1 square 6 solutions | ||||

7 | 2 squares 2 solutions | 2 squares 1 solution | 2 squares 1 solution | 1 square 2 solutions | |||||

8 | 1 square 1 solution | 1 square 1 solution | 1 square 1 solution |

Q \ A | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

2 | 3 squares 2 solutions | 12 squares 1 solution | 7 squares 3 solutions | ||||||

3 | 2 squares 2 solutions | 10 squares 1 solution | 10 squares 1 solution | 5 squares 2 solutions | |||||

4 | 1 square 25 solutions | 8 squares 1 solution | 12 squares 1 solution | 9 squares 1 solution | 4 squares 1 solution | ||||

5 | 1 square 9 solutions | 5 squares 2 solutions | 11 squares 1 solution | 10 squares 1 solutions | 6 squares 2 solutions | 2 squares 8 solutions | |||

6 | 1 square 2 solutions | 4 squares 1 solution | 8 squares 2 solutions | 10 squares 1 solution | 6 squares 17 solutions | 4 squares 2 solutions | 2 squares 2 solutions | ||

7 | 3 squares 1 solution | 6 squares 10 solutions | 9 squares 2 solutions | 7 squares 4 solutions | 5 squares 3 solutions | 2 squares 32 solutions | 1 square 29 solutions | ||

8 | 1 square 26 solutions | 6 squares 2 solutions | 8 squares 2 solutions | 8 squares 1 solution | 8 squares 1 solution | 4 squares 1 solutions | 2 squares 5 solutions | 1 square 6 solutions | |

9 | 1 square 3 solutions | 4 squares 8 solutions | 7 squares 2 solutions | 7 squares 1 solution | 6 squares 1 solution | 4 squares 2 solutions | 3 squares 1 solution | 1 square 23 solutions | |

10 | 4 squares 1 solution | 6 squares 3 solutions | 6 squares 1 solution | 5 squares 1 solution | 3 squares 12 solutions | 4 squares 1 solution | 2 squares 2 solutions | ||

11 | 3 squares 1 solution | 5 squares 2 solutions | 5 squares 1 solution | 4 squares 7 solutions | 3 squares 1 solution | 3 squares 5 solutions | 2 squares 5 solutions | ||

12 | 2 squares 2 solutions | 4 squares 3 solutions | 4 squares 2 solutions | 4 squares 4 solutions | 2 squares 5 solutions | 3 squares 1 solution | 4 squares 1 solution | ||

13 | 3 squares 2 solutions | 2 squares 10 solutions | 3 squares 4 solutions | 1 square 6 solutions | 2 squares 3 solutions | 3 squares 1 solution | |||

14 | 2 squares 2 solutions | 2 squares 1 solution | 2 squares 5 solutions | 1 square 3 solutions | 2 squares 2 solutions | ||||

15 | 1 square 1 solution | 1 square 1 solution | 1 square 1 solution |

Q \ A | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

2 | 7 squares 4 solutions | 19 squares 1 solution | 9 squares 1 solution | ||||||

3 | 5 squares 1 solution | 17 squares 1 solution | 16 squares 1 solution | 5 squares 15 solutions | |||||

4 | 4 squares 6 solutions | 15 squares 2 solutions | 18 squares 1 solution | 13 squares 1 solution | 4 squares 2 solutions | ||||

5 | 3 squares 1 solution | 14 squares 1 solution | 20 squares 1 solution | 14 squares 7 solutions | 8 squares 1 solution | 3 squares 2 solutions | |||

6 | 2 squares 38 solutions | 12 squares 2 solutions | 17 squares 2 solutions | 17 squares 1 solution | 9 squares 8 solutions | 4 squares 7 solutions | 2 squares 8 solutions | ||

7 | 2 squares 4 solutions | 10 squares 1 solution | 15 squares 2 solutions | 17 squares 2 solutions | 12 squares 8 solutions | 6 squares 14 solutions | 2 squares 437 solutions | 1 square 367 solutions | |

8 | 1 square 1056 solutions | 8 squares 1 solution | 13 squares 9 solutions | 16 squares 3 solutions | 14 squares 1 solution | 8 squares 1 solution | 5 squares 1 solution | 2 squares 36 solutions | 1 square 99 solutions |

9 | 1 square 450 solutions | 6 squares 2 solutions | 13 squares 1 solution | 16 squares 2 solutions | 16 squares 1 solution | 9 squares 6 solutions | 6 squares 4 solutions | 4 squares 1 solution | 1 square 898 solutions |

10 | 1 square 140 solutions | 5 squares 1 solution | 10 squares 14 solutions | 15 squares 1 solution | 15 squares 1 solution | 11 squares 4 solutions | 7 squares 1 solution | 4 squares 10 solutions | 2 squares 20 solutions |

11 | 1 square 24 solutions | 4 squares 1 solution | 10 squares 1 solution | 14 squares 2 solutions | 14 squares 2 solutions | 12 squares 2 solutions | 7 squares 20 solutions | 5 squares 17 solutions | 2 squares 248 solutions |

12 | 1 square 3 solutions | 3 squares 9 solutions | 9 squares 1 solution | 13 squares 2 solutions | 13 squares 2 solutions | 11 squares 9 solutions | 8 squares 5 solutions | 7 squares 2 solutions | 4 squares 3 solutions |

13 | 3 squares 1 solution | 8 squares 16 solutions | 12 squares 2 solutions | 12 squares 2 solutions | 12 squares 2 solutions | 8 squares 6 solutions | 7 squares 10 solutions | 4 squares 20 solutions | |

14 | 1 square 404 solutions | 8 squares 1 solution | 11 squares 2 solutions | 11 squares 2 solutions | 11 squares 1 solution | 7 squares 15 solutions | 7 squares 21 solutions | 6 squares 2 solutions | |

15 | 1 square 54 solutions | 6 squares 2 solutions | 10 squares 3 solutions | 10 squares 3 solutions | 10 squares 2 solutions | 6 squares 56 solutions | 7 squares 11 solutions | 6 squares 9 solutions | |

16 | 1 square 4 solutions | 5 squares 1 solution | 9 squares 3 solutions | 9 squares 3 solutions | 9 squares 4 solutions | 6 squares 1 solution | 6 squares 99 solutions | 9 squares 1 solution | |

17 | 4 squares 13 solutions | 8 squares 4 solutions | 8 squares 2 solutions | 8 squares 14 solutions | 5 squares 7 solutions | 6 squares 14 solutions | 8 squares 3 solutions | ||

18 | 4 squares 1 solution | 7 squares 4 solutions | 7 squares 2 solutions | 7 squares 28 solutions | 4 squares 12 solutions | 5 squares 54 solutions | 7 squares 8 solutions | ||

19 | 3 squares 1 solution | 6 squares 4 solutions | 6 squares 1 solution | 6 squares 47 solutions | 3 squares 58 solutions | 5 squares 2 solutions | 6 squares 16 solutions | ||

20 | 2 squares 2 solutions | 5 squares 2 solutions | 5 squares 1 solution | 5 squares 53 solutions | 3 squares 5 solutions | 4 squares 4 solutions | 5 squares 23 solutions | ||

21 | 4 squares 3 solutions | 4 squares 2 solutions | 4 squares 48 solutions | 2 squares 4 solutions | 3 squares 7 solutions | 4 squares 23 solutions | |||

22 | 3 squares 2 solutions | 2 squares 15 solutions | 3 squares 25 solutions | 1 square 8 solutions | 2 squares 7 solutions | 3 squares 16 solutions | |||

23 | 2 squares 2 solutions | 2 squares 1 solution | 2 squares 11 solutions | 1 square 5 solutions | 2 squares 8 solutions | ||||

24 | 1 square 1 solution | 1 square 2 solutions | 1 square 3 solutions |

Q \ A | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

2 | 13 squares 3 solutions | 24 squares 1 solution | 10 squares 3 solutions | ||||||

3 | 9 squares 9 solutions | 26 squares 1 solution | 19 squares 2 solutions | 5 squares 34 solutions | |||||

4 | 8 squares 2 solutions | 24 squares 1 solution | 24 squares 1 solution | 14 squares 1 solution | 4 squares 5 solutions | ||||

5 | 6 squares 3 solutions | 23 squares 1 solution | 24 squares 3 solutions | 18 squares 3 solutions | 8 squares 1 solution | 3 squares 4 solutions | |||

6 | 5 squares 3 solutions | 21 squares 3 solutions | 28 squares 2 solutions | 22 squares 4 solutions | 12 squares 5 solutions | 4 squares 27 solutions | 2 squares 24 solutions | ||

7 | 4 squares 13 solutions (GS) | 20 squares 1 solution (GS) | 23 squares 9 solutions (GS) | 24 squares 2 solutions (GS) | 16 squares 2 solutions (GS) | 7 squares 4 solutions (GS) | 2 squares 2819 solutions (GS) | 1 square 2444 solutions (GS) | |

8 | 4 squares 2 solutions (GS) | 18 squares 1 solution (GS) | 23 squares 1 solution (GS) | 25 squares 1 solution (GS) | 24 squares 1 solution (GS) | 10 squares 2 solutions (GS) | 5 squares 2 solutions (GS) | 2 squares 213 solutions (GS) | 1 square 652 solutions (GS) |

9 | 3 squares 9 solutions (GS) | 17 squares 1 solution (GS) | 21 squares 2 solutions (GS) | 26 squares 1 solution (GS) | 24 squares 1 solution (GS) | 12 squares 1 solution (GS) | 6 squares 27 solutions (GS) | 4 squares 1 solution (GS) | 1 square 13314 solutions (GS) |

10 | 2 squares 1736 solutions (GS) | 14 squares 2 solutions (GS) | 20 squares 1 solution (GS) | 25 squares 1 solution (GS) | 23 squares 2 solutions (GS) | 14 squares 2 solutions (GS) | 8 squares 9 solutions (GS) | 4 squares 65 solutions (GS) | 2 squares 157 solutions (GS) |

11 | 2 squares 379 solutions (GS) | 12 squares 2 solutions (GS) | 18 squares 1 solution (GS) | 25 squares 1 solution (GS) | 23 squares 1 solution (GS) | 16 squares 2 solutions (GS) | 10 squares 2 solutions (GS) | 5 squares 2 solutions (GS) | 2 squares 4442 solutions (GS) |

12 | 2 squares 53 solutions (GS) | 10 squares 2 solutions (GS) | 17 squares 2 solutions (GS) | 24 squares 1 solution (GS) | 24 squares 2 solutions (GS) | 19 squares 2 solutions (GS) | 12 squares 1 solution (GS) | 6 squares 2 solutions (GS) | 2 squares 15 solutions (GS) |

13 | 2 squares 3 solutions (GS) | 8 squares 5 solutions (GS) | 15 squares 59 solutions (GS) | 23 squares 1 solution (GS) | 23 squares 2 solutions (GS) | 19 squares 6 solutions (GS) | 12 squares 6 solutions (GS) | 8 squares 35 solutions (GS) | 4 squares 282 solutions (GS) |

14 | 1 square 101632 solutions (GS) | 7 squares 2 solutions (GS) | 15 squares 5 solutions (GS) | 22 squares 2 solutions (GS) | 22 squares 6 solutions (GS) | 20 squares 3 solutions (GS) | 13 squares 7 solutions (GS) | 11 squares 1 solution (GS) | 6 squares 9 solutions (GS) |

15 | 1 square 39992 solutions (GS) | 6 squares 4 solutions (GS) | 14 squares 10 solutions (GS) | 21 squares 2 solutions (GS) | 21 squares 4 solutions (GS) | 20 squares 2 solutions (GS) | 13 squares 19 solutions (GS) | 12 squares 2 solutions (GS) | 6 squares 173 solutions (GS) |

16 | 1 square 12370 solutions (GS) | 5 squares 16 solutions (GS) | 14 squares 1 solution (GS) | 20 squares 1 solution (GS) | 20 squares 5 solutions (GS) | 20 squares 9 solutions (GS) | 14 squares 1 solution (GS) | 12 squares 34 solutions (GS) | 9 squares 2 solutions (GS) |

17 | 1 square 2877 solutions (GS) | 5 squares 1 solution (GS) | 13 squares 1 solution (GS) | 18 squares 25 solutions (GS) | 19 squares 5 solutions (GS) | 19 squares 3 solutions (GS) | 13 squares 22 solutions (GS) | 13 squares 4 solutions (GS) | 9 squares 23 solutions (GS) |

18 | 1 square 482 solutions (GS) | 4 squares 2 solutions (GS) | 12 squares 2 solutions (GS) | 18 squares 3 solutions (GS) | 18 squares 5 solutions (GS) | 18 squares 1 solution (GS) | 13 squares 3 solutions (GS) | 13 squares 6 solutions (GS) | 12 squares 2 solutions (GS) |

19 | 1 square 50 solutions (GS) | 3 squares 128 solutions (GS) | 11 squares 3 solutions (GS) | 17 squares 1 solution (GS) | 17 squares 6 solutions (GS) | 17 squares 2 solutions (GS) | 12 squares 26 solutions (GS) | 13 squares 2 solutions (GS) | 12 squares 10 solutions (GS) |

20 | 1 square 3 solutions (GS) | 3 squares 4 solutions (GS) | 10 squares 143 solutions (GS) | 16 squares 3 solutions (GS) | 16 squares 4 solutions (GS) | 16 squares 9 solutions (GS) | 12 squares 1 solution (GS) | 12 squares 82 solutions (GS) | 16 squares 1 solution (GS) |

21 | 3 squares 1 solution (GS) | 10 squares 12 solutions (GS) | 15 squares 4 solutions (GS) | 15 squares 3 solutions (GS) | 15 squares 16 solutions (GS) | 11 squares 2 solutions (GS) | 12 squares 5 solutions (GS) | 15 squares 3 solutions (GS) | |

22 | 1 squares 8683 solution (GS) | 10 squares 1 solution (GS) | 14 squares 2 solutions (GS) | 14 squares 5 solutions (GS) | 14 squares 49 solutions (GS) | 10 squares 5 solutions (GS) | 11 squares 67 solutions (GS) | 14 squares 21 solutions (GS) | |

23 | 1 squares 1110 solution (GS) | 8 squares 1 solution (GS) | 13 squares 5 solutions (GS) | 13 squares 2 solutions (GS) | 13 squares 113 solutions (GS) | 9 squares 7 solutions (GS) | 11 squares 1 solution (GS) | 13 squares 77 solutions (GS) | |

24 | 1 squares 92 solution (GS) | 6 squares 3 solutions (GS) | 12 squares 6 solutions (GS) | 12 squares 4 solutions (GS) | 12 squares 249 solutions (GS) | 8 squares 21 solutions (GS) | 10 squares 8 solutions (GS) | 12 squares 252 solutions (GS) | |

25 | 1 squares 4 solution (GS) | 5 squares 28 solutions (GS) | 11 squares 4 solutions (GS) | 11 squares 2 solutions (GS) | 11 squares 413 solutions (GS) | 7 squares 79 solutions (GS) | 9 squares 40 solutions (GS) | 11 squares 567 solutions (GS) | |

26 | 5 squares 1 solution (GS) | 10 squares 5 solutions (GS) | 10 squares 3 solutions (GS) | 10 squares 630 solutions (GS) | 7 squares 2 solutions (GS) | 8 squares 151 solutions (GS) | 10 squares 1051 solutions (GS) | ||

27 | 4 squares 16 solutions (GS) | 9 squares 4 solutions (GS) | 9 squares 2 solutions (GS) | 9 squares 777 solutions (GS) | 6 squares 4 solutions (GS) | 7 squares 330 solutions (GS) | 9 squares 1465 solutions (GS) | ||

28 | 4 squares 1 solution (GS) | 8 squares 4 solutions (GS) | 8 squares 1 solution (GS) | 8 squares 866 solutions (GS) | 5 squares 7 solutions (GS) | 7 squares 2 solutions (GS) | 8 squares 1674 solutions (GS) | ||

29 | 3 squares 1 solution (GS) | 7 squares 4 solutions (GS) | 7 squares 2 solutions (GS) | 7 squares 784 solutions (GS) | 4 squares 49 solutions (GS) | 6 squares 5 solutions (GS) | 7 squares 1465 solutions (GS) | ||

30 | 2 squares 2 solutions (GS) | 6 squares 4 solutions (GS) | 6 squares 1 solution (GS) | 6 squares 634 solutions (GS) | 4 squares 3 solutions (GS) | 5 squares 8 solutions (GS) | 6 squares 1051 solutions (GS) | ||

31 | 5 squares 2 solutions (GS) | 5 squares 1 solution (GS) | 5 squares 385 solutions (GS) | 3 squares 3 solutions (GS) | 4 squares 11 solutions (GS) | 5 squares 567 solutions (GS) | |||

32 | 4 squares 3 solutions (GS) | 4 squares 2 solutions (GS) | 4 squares 198 solutions (GS) | 2 squares 5 solutions (GS) | 3 squares 11 solutions (GS) | 4 squares 252 solutions (GS) | |||

33 | 3 squares 2 solutions (GS) | 2 squares 21 solutions (GS) | 3 squares 63 solutions (GS) | 1 squares 9 solutions (GS) | 2 squares 9 solutions (GS) | 3 squares 77 solutions (GS) | |||

34 | 2 squares 2 solutions (GS) | 2 squares 1 solution (GS) | 2 squares 19 solutions (GS) | 1 squares 6 solutions (GS) | 2 squares 21 solutions (GS) | ||||

35 | 1 square 1 solution (GS) | 1 square 2 solutions (GS) | 1 square 3 solutions (GS) |

Q \ A | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

2 | 21 squares 6 solutions | 31 squares 1 solutions | 11 squares 2 solutions | ||||||

3 | 16 squares 5 solutions | 34 squares 1 solution | 22 squares 3 solutions | 5 squares 77 solutions | |||||

4 | 16 squares 1 solution | 34 squares 1 solution | 29 squares 1 solution | 14 squares 3 solutions | 4 squares 11 solutions | ||||

5 | 11 squares 1 solution | 32 squares 5 solutions | 33 squares 1 solution | 21 squares 2 solutions | 8 squares 4 solutions | 3 squares 11 solutions | |||

6 | 10 squares 1 solution (GS) | 32 squares 1 solution (GS) | 34 squares 5 solutions (GS) | 28 squares 1 solution (GS) | 12 squares 16 solutions (GS) | 4 squares 65 solutions (GS) | 2 squares 69 solutions (GS) | ||

7 | 7 squares 38 solutions (GS) | 30 squares 7 solutions (GS) | 38 squares 2 solutions (GS) | 33 squares 1 solution (GS) | 16 squares 31 solutions (GS) | 7 squares 18 solutions (GS) | 2 squares 12279 solutions (GS) | 1 square 12508 solutions (GS) | |

8 | 6 squares 174 solutions (GS) | 30 squares 1 solution (GS) | 40 squares 1 solution (GS) | 33 squares 1 solution (GS) | 25 squares 1 solution (GS) | 10 squares 9 solutions (GS) | 5 squares 7 solutions (GS) | 2 squares 831 solutions (GS) | 1 square 4030 solutions (GS) |

9 | 6 squares 19 solutions (GS) | 28 squares 1 solution (GS) | 40 squares 1 solution (GS) | 35 squares 1 solution (GS) | 29 squares 1 solution (GS) | 12 squares 47 solutions (GS) | 6 squares 102 solutions (GS) | 4 squares 4 solutions (GS) | 1 square 123279 solutions (GS) |

10 | 6 squares 1 solution (GS) | 26 squares 2 solutions (GS) | 29 squares 5 solutions (GS) | 36 squares 1 solution (GS) | 31 squares 2 solutions (GS) | 16 squares 3 solutions (GS) | 9 squares 4 solutions (GS) | 4 squares 256 solutions (GS) | 2 squares 909 solutions (GS) |

Q \ A | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

2 | 31 squares 6 solutions | 38 squares 1 solution | 12 squares 2 solutions | ||||||

3 | 25 squares 3 solutions | 43 squares 3 solutions | 23 squares 5 solutions | 5 squares 124 solutions | |||||

4 | 24 squares 1 solution | 46 squares 1 solution | 36 squares 1 solution | 14 squares 7 solutions | 4 squares 22 solutions | ||||

5 | 17 squares 2 solutions | 43 squares 3 solutions | 39 squares 2 solutions | 22 squares 5 solutions | 8 squares 5 solutions | 3 squares 16 solutions | |||

6 | 15 squares 2 solutions (GS) | 43 squares 1 solution (GS) | 34 squares 1 solution (GS) | 31 squares 1 solution (GS) | 12 squares 36 solutions (GS) | 4 squares 133 solutions (GS) | 2 squares 138 solutions (GS) | ||

7 | 13 squares 1 solution (GS) | 41 squares 11 solutions (GS) | 44 squares 3 solutions (GS) | 38 squares 1 solution (GS) | 17 squares 4 solutions (GS) | 7 squares 47 solutions (GS) | 2 squares 40573 solutions (GS) | 1 square 48799 solutions (GS) | |

8 | 11 squares 7 solutions (GS) | 41 squares 2 solutions (GS) | 48 squares 4 solutions (GS) | 41 squares 9 solutions (GS) | 25 squares 1 solution (GS) | 10 squares 33 solutions (GS) | 5 squares 11 solutions (GS) | 2 squares 2655 solutions (GS) | 1 square 17442 solutions (GS) |

Q \ A | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

2 | 43 squares 10 solutions | 45 squares 1 solution | 12 squares 4 solutions | |||||

3 | 36 squares 4 solutions | 54 squares 3 solutions | 25 squares 6 solutions | 5 squares 219 solutions | ||||

4 | 36 squares 1 solution | 56 squares 2 solutions | 40 squares 1 solution | 14 squares 12 solutions | 4 squares 36 solutions | |||

5 | 27 squares 1 solution (GS) | 56 squares 5 solutions (GS) | 52 squares 1 solution (GS) | 23 squares 4 solutions (GS) | 8 squares 11 solutions (GS) | 3 squares 31 solutions (GS) | ||

6 | 24 squares 1 solution (GS) | 55 squares 1 solution (GS) | 52 squares 1 solution (GS) | 33 squares 2 solutions (GS) | 12 squares 66 solutions (GS) | 4 squares 232 solutions (GS) | 2 squares 299 solutions (GS) | |

7 | 20 squares 5 solutions (GS) | 53 squares 19 solutions (GS) | 53 squares 3 solutions (GS) | 42 squares 4 solutions (GS) | 18 squares 8 solutions (GS) | 7 squares 97 solutions (GS) | 2 squares 112572 solutions (GS) | 1 square 161826 solutions (GS) |

Q \ A | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

2 | 57 squares 10 solutions | 52 squares 1 solution | 12 squares 8 solutions | |||||

3 | 49 squares 3 solutions | 65 squares 2 solutions | 26 squares 11 solutions | 5 squares 316 solutions | ||||

4 | 48 squares 1 solution (GS) | 72 squares 1 solution (GS) | 48 squares 1 solution (GS) | 14 squares 20 solutions (GS) | 4 squares 60 solutions (GS) | |||

5 | 37 squares 1 solution (GS) | 74 squares 1 solution (GS) | 56 squares 2 solutions (GS) | 23 squares 10 solutions (GS) | 8 squares 14 solutions (GS) | 3 squares 40 solutions (GS) | ||

6 | 33 squares 1 solution (GS) | 72 squares 1 solution (GS) | 64 squares 1 solution (GS) | 34 squares 3 solutions (GS) | 12 squares 110 solutions (GS) | 4 squares 370 solutions (GS) | 2 squares 497 solutions (GS) | |

7 | 30 squares 2 solutions (GS) | 67 squares 7 solutions (GS) | 65 squares 1 solution (GS) | 44 squares 7 solutions (GS) | 18 squares 20 solutions (GS) | 7 squares 164 solutions (GS) | 2 squares 272297 solutions (GS) | 1 square 462795 solutions (GS) |

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