Calculation Challenge Game:

Some Statistics

All of the statistics on this page are specific to the Calculation Challenge Game when the Goal integer = 24 and the factor range is 1 to 13.

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As the table above shows, there are more valid solutions if one of the factors is 1, 2, 3, 4, 6, 8, or 12 and fewer valid solutions if one of the factors is 5, 7, 9, 10, 11, or 13, which seems pretty reasonable.

Note:  In the table above, there are a total of 1276248 + 217089120 + 983112 = 219348480 possible equations with 4 factors, in which each of the 4 factors can be 1 to 13.  We can verify this total by considering how many possible equations we would expect:  There are 13 possible numbers for each of the 4 factors and 7680 ways to combine them using 4 operations (+, -, x, /) and 2 sets of parentheses, which gives 13⁴ x 7680 = 219348480 expected equations.  (We’ve calculated the number of possible equations, not the number of unique equations!)

 

 

Unsolvable factor sets

There are 459 unique unsolvable factor sets (listed here).  These are sets of factors in which no combination of adding, subtracting, multiplying, or dividing the 4 factors results in 24.  The table below shows the number of unsolvable sets that contain at least one of the factors from 1 to 13:

As the table above shows, over a third of the unsolvable sets contain at least one factor equal to 13.  Additionally, over a quarter of the unsolvable sets contain a factor of 9, 10, 7, or 11, with the factor of 1 very nearly in that category.  At the other extreme, only 13% of the unsolvable sets contain a 2 or a 3.

We state elsewhere that the largest number of valid solutions occurs when the factors are all 12s.  In general though, having all the same factor is more likely to be an unsolvable set of factors.  Of the possible 13 factor sets with all the same factors, only 5 of them have valid solutions (those in which the factors are all 3, 4, 5, 6, or 12); the remaining 8 factor sets are unsolvable (those in which the factors are all 1, 2, 7, 8, 9, 10, 11, or 13).

So, how many unique combinations of 4 numbers are there in which each number can be 1 to 13?

There are 5 cases to consider:

                 Case 1:  None of the 4 numbers are repeated

                                  13 x 12 x 11 x 10 = 17160

                 Case 2:  1 number is repeated and the other 2 numbers are different

                                  (13 x 1) x 12 x 11 = 1716

                 Case 3:  2 numbers are repeated

                                  (13 x 1) x (12 x 1) = 156

                 Case 4:  1 number is repeated 3 times and the 4th number is different

                                  (13 x 1 x 1) x 12 = 156

                 Case 5:  1 number is repeated 4 times

                                  (13 x 1 x 1 x 1) = 13

Adding up these 5 cases, we calculate that there are 17160 + 1716 + 156 + 156 + 13 = 19201 unique factor sets of 4 numbers.

We can use the above results to calculate that 459 / 19201 = 2.39% of the factor sets are unsolvable, which means that 97.61% are solvable.  In other words, don’t be in a hurry to give up on finding a solution—nearly all of the factor sets are solvable!