The N Queens Puzzle

List of First Solutions

This page lists the first possible solutions to the n Queens on an n by n chessboard puzzle for various sizes of chessboard, as well as the numbers of Queens it is necessary to place whilst seeking those first solutions using our algorithm. With two notable exceptions, the final Queen to be placed completes the first solution. The two exceptions are chessboard sizes 2 by 2 and 3 by 3, for which no solutions exist. Their "Number of Queens placed" values below are therefore the total numbers of Queen placements needed to search the entire chessboard.

All of the solutions and Queen placement counts have been generated by a simple program written by my Dad, Colin using Borland C++. This is in fact the latest in a long line of similar programs that he's written to work on this puzzle over the years, beginning way back in 1976 using Assembly Language on a big, slow old mainframe computer! The results for all chessboard sizes up to 39 by 39 have been verified by a program I wrote independently in July 2008 using Borland Turbo Pascal 7.01 (DOS Version), and I hope to verify the results for the higher chessboard sizes in due course. Further results will be added to this page as they are generated.

We have found surprisingly few consistent patterns in the numbers of Queens placed whilst seeking the first solutions. As one might expect, there is a general trend for the numbers to increase along with the chessboard sizes, but within this trend the numbers actually fluctuate up and down quite considerably. The only consistent patterns we've identified so far are as follows:

• Each even-numbered chessboard requires more Queens to be placed on it whilst seeking the first solution than the odd-numbered chessboard immediately before it. For example, the total for 8 by 8 is 876, which is noticeably higher than the total for 7 by 7 (42).
• The total for an odd-numbered chessboard is always exactly divisible by the chessboard size. For example, the total for 7 by 7 is 42, which when divided by 7 gives 6.
• The total for an even-numbered chessboard is always exactly divisible by half the chessboard size, but it is NOT exactly divisible by the chessboard size itself. For example, the total for 6 by 6 is 171, which when divided by 3 gives 57, but when divided by 6 gives 28.5.

The numbers of Queens placed whilst seeking the first solutions were accepted into The On-Line Encyclopedia of Integer Sequences (OEIS) on 26th June 2008 as sequence A140450. In addition, the first solutions themselves were accepted into the OEIS on 10th July 2008 as sequence A141843.

So without further ado, here are the results of all our hard work, given in our simple co-ordinate notation. Click on the "first solution" links to see chessboard diagrams for each size of chessboard (except for 2 by 2 and 3 by 3, of course).

Chessboard Size 1 by 1

• Number of Queens placed whilst seeking the first solution: 1
• The one "solution" for this size of chessboard is of course just a single Queen!
• The first solution described using our simple co-ordinate notation:
  1

Chessboard Size 2 by 2

• Number of Queens placed whilst seeking the first solution: 6
• No solutions exist for this size of chessboard.

Chessboard Size 3 by 3

• Number of Queens placed whilst seeking the first solution: 18
• No solutions exist for this size of chessboard.

Chessboard Size 4 by 4

• Number of Queens placed whilst seeking the first solution: 26
• The first solution described using our simple co-ordinate notation:
  2, 4, 1, 3

Chessboard Size 5 by 5

• Number of Queens placed whilst seeking the first solution: 15
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4

Chessboard Size 6 by 6

• Number of Queens placed whilst seeking the first solution: 171
• The first solution described using our simple co-ordinate notation:
  2, 4, 6, 1, 3, 5

Chessboard Size 7 by 7

• Number of Queens placed whilst seeking the first solution: 42
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 7, 2, 4, 6

Chessboard Size 8 by 8

• Number of Queens placed whilst seeking the first solution: 876
• The first solution described using our simple co-ordinate notation:
  1, 5, 8, 6, 3, 7, 2, 4

Chessboard Size 9 by 9

• Number of Queens placed whilst seeking the first solution: 333
• The first solution described using our simple co-ordinate notation:
  1, 3, 6, 8, 2, 4, 9, 7, 5

Chessboard Size 10 by 10

• Number of Queens placed whilst seeking the first solution: 975
• The first solution described using our simple co-ordinate notation:
  1, 3, 6, 8, 10, 5, 9, 2, 4, 7

Chessboard Size 11 by 11

• Number of Queens placed whilst seeking the first solution: 516
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 7, 9, 11, 2, 4, 6, 8, 10

Chessboard Size 12 by 12

• Number of Queens placed whilst seeking the first solution: 3,066
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 8, 10, 12, 6, 11, 2, 7, 6, 4

Chessboard Size 13 by 13

• Number of Queens placed whilst seeking the first solution: 1,365
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 9, 12, 10, 13, 4, 6, 8, 11, 7

Chessboard Size 14 by 14

• Number of Queens placed whilst seeking the first solution: 26,495
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 7, 12, 10, 13, 4, 14, 9, 2, 6, 8, 11

Chessboard Size 15 by 15

• Number of Queens placed whilst seeking the first solution: 20,280
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 10, 12, 14, 4, 13, 9, 6, 15, 7, 11, 8

Chessboard Size 16 by 16

• Number of Queens placed whilst seeking the first solution: 160,712
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 13, 9, 14, 12, 15, 6, 16, 7, 4, 11, 8, 10

Chessboard Size 17 by 17

• Number of Queens placed whilst seeking the first solution: 91,222
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 8, 11, 15, 7, 16, 14, 17, 4, 6, 9, 12, 10, 13

Chessboard Size 18 by 18

• Number of Queens placed whilst seeking the first solution: 743,229
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 8, 15, 12, 16, 13, 17, 6, 18, 7, 4, 11, 9, 14, 10

Chessboard Size 19 by 19

• Number of Queens placed whilst seeking the first solution: 48,184
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 13, 15, 17, 19, 7, 16, 18, 11, 6, 8, 10, 12, 14

Chessboard Size 20 by 20

• Number of Queens placed whilst seeking the first solution: 3,992,510
• This is the first chessboard to need more than one million Queen placements.
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 13, 15, 12, 18, 20, 17, 9, 16, 19, 8, 10, 7, 14, 6, 11

Chessboard Size 21 by 21

• Number of Queens placed whilst seeking the first solution: 179,592
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 15, 21, 18, 20, 17, 19, 7, 12, 10, 8, 6, 14, 16, 13

Chessboard Size 22 by 22

• Number of Queens placed whilst seeking the first solution: 38,217,905
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 10, 14, 17, 20, 13, 19, 22, 18, 8, 21, 12, 9, 6, 16, 7, 11, 15

Chessboard Size 23 by 23

• Number of Queens placed whilst seeking the first solution: 584,591
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 18, 20, 22, 19, 21, 10, 8, 6, 23, 7, 16, 12, 15, 17, 14

Chessboard Size 24 by 24

• Number of Queens placed whilst seeking the first solution: 9,878,316
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 14, 18, 22, 19, 23, 20, 24, 10, 21, 6, 8, 12, 16, 13, 7, 17, 15

Chessboard Size 25 by 25

• Number of Queens placed whilst seeking the first solution: 1,216,775
• This is the first odd-numbered chessboard to need more than one million Queen placements.
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 19, 21, 24, 20, 25, 23, 6, 8, 10, 7, 14, 16, 18, 12, 17, 22

Chessboard Size 26 by 26

• Number of Queens placed whilst seeking the first solution: 10,339,849
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 21, 23, 25, 20, 22, 24, 26, 10, 7, 16, 12, 8, 6, 18, 14, 19, 17

Chessboard Size 27 by 27

• Number of Queens placed whilst seeking the first solution: 12,263,400
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 17, 19, 23, 25, 27, 24, 26, 6, 10, 7, 16, 8, 12, 14, 21, 18, 20, 22

Chessboard Size 28 by 28

• Number of Queens placed whilst seeking the first solution: 84,175,966
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 17, 23, 25, 22, 28, 26, 24, 27, 7, 12, 16, 18, 8, 10, 14, 20, 6, 21, 19

Chessboard Size 29 by 29

• Number of Queens placed whilst seeking the first solution: 44,434,525
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 20, 24, 26, 21, 29, 27, 25, 28, 8, 12, 7, 16, 10, 17, 22, 14, 18, 23, 19

Chessboard Size 30 by 30

• Number of Queens placed whilst seeking the first solution: 1,692,888,135
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 7, 23, 26, 28, 25, 22, 24, 30, 27, 29, 16, 12, 10, 8, 6, 18, 20, 17, 14, 21, 19

Chessboard Size 31 by 31

• Number of Queens placed whilst seeking the first solution: 408,773,285
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 18, 23, 26, 28, 31, 25, 27, 30, 7, 17, 29, 14, 10, 8, 20, 12, 16, 19, 22, 24, 21

Chessboard Size 32 by 32

• Number of Queens placed whilst seeking the first solution: 2,799,725,104
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 18, 24, 26, 30, 25, 31, 28, 32, 27, 29, 16, 19, 10, 8, 17, 12, 21, 7, 14, 23, 20, 22

Chessboard Size 33 by 33

• Number of Queens placed whilst seeking the first solution: 4,618,568,460
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 25, 27, 33, 31, 23, 28, 26, 29, 32, 30, 16, 18, 12, 10, 17, 7, 14, 21, 19, 24, 22, 20

Chessboard Size 34 by 34

• Number of Queens placed whilst seeking the first solution: 78,016,579,095
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 18, 20, 26, 28, 31, 33, 27, 29, 34, 32, 30, 12, 10, 7, 16, 19, 8, 22, 14, 25, 17, 24, 21, 23

Chessboard Size 35 by 35

• Number of Queens placed whilst seeking the first solution: 8,244,234,495
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 24, 26, 31, 29, 32, 35, 33, 28, 30, 17, 7, 34, 10, 18, 16, 14, 12, 20, 25, 23, 21, 27, 22

Chessboard Size 36 by 36

• Number of Queens placed whilst seeking the first solution: 883,134,227,754
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 22, 27, 32, 30, 33, 24, 29, 35, 28, 34, 31, 14, 36, 17, 12, 16, 7, 10, 20, 25, 19, 26, 18, 23, 21

Chessboard Size 37 by 37

• Number of Queens placed whilst seeking the first solution: 37,481,487,475
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 21, 23, 30, 34, 29, 35, 33, 31, 37, 32, 36, 16, 7, 10, 20, 14, 12, 18, 26, 17, 22, 28, 25, 27, 24

Chessboard Size 38 by 38

• Number of Queens placed whilst seeking the first solution: 605,847,021,365
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 21, 29, 31, 33, 36, 30, 28, 38, 34, 32, 37, 35, 20, 18, 12, 14, 23, 7, 10, 25, 17, 22, 16, 26, 24, 27

Chessboard Size 39 by 39

• Number of Queens placed whilst seeking the first solution: 444,710,580,405
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 7, 25, 30, 37, 33, 28, 32, 38, 35, 39, 31, 36, 34, 21, 17, 20, 14, 12, 10, 23, 16, 18, 22, 29, 26, 24, 27

The results for all sizes of chessboard below this point have not yet been independently verified.

Chessboard Size 40 by 40

• Number of Queens placed whilst seeking the first solution: 18,325,192,478,100
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 21, 23, 30, 32, 34, 36, 38, 40, 31, 33, 35, 37, 39, 20, 22, 14, 18, 10, 12, 24, 17, 7, 28, 26, 16, 29, 27, 25

Chessboard Size 41 by 41

• Number of Queens placed whilst seeking the first solution: 670,447,199,438
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 7, 22, 25, 30, 32, 37, 39, 33, 38, 40, 34, 36, 41, 35, 18, 23, 10, 12, 16, 20, 17, 14, 21, 28, 26, 31, 29, 27, 24

Chessboard Size 42 by 42

• Number of Queens placed whilst seeking the first solution: 1,435,630,462,823,151
• This is the highest number of Queens placed so far, with a massive sixteen digits!
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 21, 23, 25, 32, 34, 36, 38, 40, 42, 33, 35, 37, 39, 41, 16, 7, 24, 20, 14, 17, 26, 10, 12, 30, 28, 18, 31, 22, 27, 29

Chessboard Size 43 by 43

• Number of Queens placed whilst seeking the first solution: 9,387,957,604,049
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 7, 22, 24, 26, 37, 39, 34, 38, 35, 33, 42, 36, 41, 43, 40, 20, 14, 17, 10, 12, 28, 18, 16, 27, 25, 21, 32, 30, 23, 31, 29

Chessboard Size 44 by 44

• Number of Queens placed whilst seeking the first solution: 168,220,196,252,658
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 7, 22, 24, 32, 35, 38, 41, 33, 42, 37, 34, 43, 39, 36, 44, 40, 21, 25, 18, 14, 12, 20, 16, 29, 10, 23, 27, 30, 17, 31, 26, 28

Chessboard Size 45 by 45

• Number of Queens placed whilst seeking the first solution: 151,162,963,331,040
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 7, 22, 10, 29, 31, 36, 40, 42, 32, 38, 43, 39, 44, 35, 37, 45, 41, 21, 18, 24, 14, 12, 23, 16, 20, 17, 28, 25, 34, 26, 33, 30, 27

Chessboard Size 46 by 46

• Number of Queens placed whilst seeking the first solution: More than 1,057,612,946,332,207
• Queens are still being counted for this size of chessboard.

Chessboard Size 47 by 47

• Number of Queens placed whilst seeking the first solution: 351,874,509,051,227
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 7, 22, 10, 25, 29, 36, 39, 42, 44, 33, 38, 43, 46, 40, 37, 41, 45, 47, 21, 17, 24, 26, 12, 14, 16, 18, 20, 23, 34, 28, 30, 32, 27, 35, 31

Chessboard Size 48 by 48

• Number of Queens placed whilst seeking the first solution: More than 955,828,795,808,118
• Queens are still being counted for this size of chessboard.

Chessboard Size 49 by 49

• Number of Queens placed whilst seeking the first solution: 192,114,366,691,401
• The first solution described using our simple co-ordinate notation:
  1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 7, 22, 10, 25, 27, 29, 37, 40, 45, 41, 44, 38, 48, 39, 49, 43, 46, 42, 47, 16, 12, 24, 20, 28, 17, 14, 30, 21, 18, 35, 31, 26, 23, 33, 36, 34, 32

Chessboard Size 50 by 50

• Number of Queens placed whilst seeking the first solution: More than 25,957,258,413,635
• Queens are still being counted for this size of chessboard.