Interesting Statistics about Fathom It!

Generating and solving thousands of Fathom It! boards uncovered many interesting statistics about boards, hints, possible solutions, etc.

Distribution of board difficulty ratings

All Fathom It! boards are randomly generated and then analyzed to guarantee a unique solution. This unique board configuration is then passed to the off-line solver. Based on the steps used by the solver to solve the board, the board is given one of the following difficulty ratings: easy, intermediate, hard or expert. In the event the solver cannot solve the board, it is discarded and not included in the Fathom It! board database.

The distribution of difficulty ratings for 10x10 (7x7) boards produced by the off-line solver is approximately:

 Easy:  6% (34%)

 Intermediate: 39% (59%)

 Hard:  35% (2%)

 Expert:  19% (57%)

 Unsolved: 1.5% (0.003%)

Distribution of 'starting-off' hints

To guarantee a unique solution for a board, Fathom It! provides the minimal number of 'starting-off' hints (i.e. revealed squares) to disambiguate the board. A board may have zero or more hints. The distribution of hints, taken from a sample of 32,000 randomly generated 10x10 (7x7) boards, is:

 0: 2% (5%)

 1: 7% (55%)

 2: 35% (39%)

 3: 44% (2%)

 4: 13% (0.006%)

 5: 1% (N/A)

 6: 0.003% (N/A)

To date, no 10x10 board has been found to need more than 6 hints. In a sample of 32,000 randomly generated 10x10 boards, only one board needed 6 hints.

Non-unique board solutions

Given a set of valid row and column tallies, there are, on the average, approximately 1575 solutions. Fathom It! guarantees a unique solution by revealing the minimal number of starting hints that eliminate all but one solution. The distribution of possible solutions, taken from a sample of 32,000 randomly generated 10x10 boards, is as follows:

  1-1000: 64.9%

 1001-2000: 13.9%

 2001-3000: 7.0%

 3001-4000: 4.3%

 4001-5000: 2.6%

 5001-6000: 1.8%

 6001-7000: 1.1%

 7001-above: 4.5%

The distribution of possible solutions, taken from a sample of 32,000 randomly generated 7x7 boards, is as follows:

  1 - 10: 53.0%

 11 - 20: 26.6%

 21 - 30: 11.1%

 31 - 40:  4.4%

 41 - 50:  2.3%

 51 - 60:  1.0%

 61 - above: 1.6%

Do the numbers of ambiguous solutions exhibit any mathematical properties?

It would have been quite interesting had the number of ambiguous solutions exhibited some mathematical pattern (e.g. primes, non-primes, Fibonacci series, Farey series). Examining the set of 10x10 board solutions in the range 1-1000, every number appears except for five numbers (596, 835, 838, 849 and 947). The conclusion is that a board can always be found with a specific number of ambiguous solutions.

The largest known number of solutions for any single board

To date, the largest number of solutions for a single board is 93,124.