In several recent posts, I have discussed a method for perfect solution to the game of 24s. Here, I summarize some of those results and link to the Mathematica notebook I used to find them, and a comprehensive list of solutions for those who would like to skip the math and get right to the answer.

To recap, there are 715 possible hands in a game of 24s. 566 of those hands can be solved with arithmetics alone, 595 if we add roots, powers, and logs, and 619 if we add Mod.

I generally play with Log and Power and Root allowed, but not Mod. I find that the hands that are solvable only with Log, Power, and Root, generally are solvable by humans. See for example

{1,1,8,9} -> 8 Root[9,1+1]
{1,2,9,9} -> (9-1) Root[9,2]
{2, 2, 2, 6} -> 2^(2+Log[2,6]) or (2*Root[6,2])^2
{2,2,9,9} -> 2(Root[9,2]+9)
{3,7,8,10} -> 7 Root[8,3] +10
{3,4,9,10} -> 4+Log[3,9]*10
{7, 8, 9, 9} -> 8 Root[9,9-7], and
{8, 8, 9, 10} -> 8 Root[9,10-8].

A bright high school student should be able to find any of those. However, Mod introduces solutions that would probably never be played by an unaided human being. For example,

{5, 6, 7, 10} -> 6*Mod[7^10,5], and

{5,7,7,8} -> 8*Mod[7^7,5].

There is no other solution in these two cases.

Anyway, I’ve provided a comprehensive list of solutions linked as RTF and PDF documents, and the Mathematica notebook that I used to solve the problem in the first place.

All 24s in PDF.

Again, but as an RTF file.

And, most importantly, the Mathematica notebook.

Let me know if you do anything interesting with this.

I have shown how to comprehensively solve a game of 24s that allows just arithmetic, and how to expand that to allow powers, roots, and logarithms. Now we’re going to add the modulo (Mod) function.

If you’ve followed the methodology in parts i and ii, this is pretty simple stuff.

First, to our set of simplified function definitions, I add
mod[x_Integer?(# > 0 &), y_Integer?(-16 <= # <= 16 && # != 0 &)] := Mod[x, y]; mod[x__] := Indeterminate;

Then, our rule definition step looks like
rules = generateRules[{Times, Plus, Subtract, divide, power, root, log, mod}, 4];

We're up to 512 rules here, once redundant patterns are eliminated. The complexity increases exponentially with the number of operators allowed.

The good news is that Mod makes lots of hands solvable that wouldn't be otherwise. See, for example,

{3, 5, 5, 10} -> 5*5-Mod[10,3]
{5, 6, 7, 10} -> 6*Mod[7^10,5]
{5, 7, 7, 8} -> Mod[7^7,5]*8, and
{6, 7, 7, 9} -> 6*Mod[7*7,9]

In fact, allowing Mod as an operator increases the total number of solvable hands all the way to 619 out of 715.

The unsolvable hands that remain are
{{1, 1, 1, 1}, {1, 1, 1, 2}, {1, 1, 1, 3}, {1, 1, 1, 4}, {1, 1, 1, 6}, {1, 1, 1, 7}, {1, 1, 1, 9}, {1, 1, 1, 10}, {1, 1, 2, 2}, {1, 1, 2, 3}, {1, 1, 5, 9}, {1, 1, 5, 10}, {1, 1, 6, 7}, {1, 1, 6, 10}, {1, 1, 7, 7}, {1, 1, 7, 8}, {1, 1, 7, 9}, {1, 1, 8, 10}, {1, 1, 9, 9}, {1, 1, 9, 10}, {1, 1, 10, 10}, {1, 2, 2, 2}, {1, 2, 9, 10}, {1, 2, 10, 10}, {1, 4, 9, 9}, {1, 5, 7, 7}, {1, 6, 6, 7}, {1, 6, 7, 7}, {1, 6, 7, 8}, {1, 6, 10, 10}, {1, 7, 7, 7}, {1, 7, 7, 8}, {1, 7, 10, 10}, {1, 8, 9, 9}, {1, 8, 9, 10}, {1, 8, 10, 10}, {1, 9, 9, 9}, {1, 9, 9, 10}, {1, 9, 10, 10}, {1, 10, 10, 10}, {2, 2, 2, 2}, {2, 2, 7, 9}, {2, 6, 7, 7}, {2, 7, 7, 7}, {2, 7, 7, 9}, {2, 7, 9, 9}, {2, 9, 9, 9}, {2, 9, 9, 10}, {2, 10, 10, 10}, {3, 3, 10, 10}, {3, 5, 7, 7}, {3, 10, 10, 10}, {4, 4, 9, 9}, {4, 7, 7, 9}, {4, 7, 7, 10}, {4, 9, 9, 9}, {5, 5, 5, 7}, {5, 5, 5, 8}, {5, 5, 5, 10}, {5, 5, 6, 10}, {5, 5, 7, 9}, {5, 7, 7, 7}, {5, 7, 9, 9}, {5, 8, 10, 10}, {5, 9, 9, 9}, {5, 10, 10, 10}, {6, 6, 6, 7}, {6, 6, 7, 7}, {6, 6, 7, 8}, {6, 7, 7, 7}, {6, 7, 7, 8}, {6, 9, 10, 10}, {7, 7, 7, 7}, {7, 7, 7, 8}, {7, 7, 7, 9}, {7, 7, 7, 10}, {7, 7, 8, 8}, {7, 7, 8, 9}, {7, 7, 9, 9}, {7, 8, 8, 8}, {7, 9, 9, 9}, {7, 9, 9, 10}, {7, 9, 10, 10}, {7, 10, 10, 10}, {8, 8, 8, 8}, {8, 8, 9, 9}, {8, 8, 10, 10}, {8, 9, 9, 9}, {8, 9, 9, 10}, {8, 9, 10, 10}, {8, 10, 10, 10}, {9, 9, 9, 9}, {9, 9, 9, 10}, {9, 9, 10, 10}, {9, 10, 10, 10}, {10, 10, 10, 10}}

In my last post on this topic, I showed that 566 of 715 possible hands in the game of 24s could be solved with simple arithmetic. However in practice people don’t play with just those operators. In my experience, a normal game always includes powers, roots, and logarithms. The solution method used for arithmetics works just fine when expanded for these additional functions, though I chose to make some small modifications for the sake of efficiency.

First, I redefine the functions to remove some cases that are quite unlikely to produce 24s, and which would increase memory needs and processing time significantly if left in place. See the following, for example:


Off[General::"spell1"]; Off[General::"spell"];
power[x_?(# > 0 &), y_?(-16 <= # <= 16 &)] := x^y; power[x__] := Indeterminate; root[x_Integer?(# > 0 &), y_Integer?(-16 <= # <= 16 && # != 0 &)] := Power[x, (y)^-1]; root[x__] := Indeterminate; divide[x_, y_] /; y != 0 := Divide[x, y]; divide[x_, 0] := Indeterminate; log[x_?(# > 0 &), y_?(# > 0 &)] := Log[x, y];
log[x__] := Indeterminate;


We can then generate rules as in the arithmetic case but with

rules = generateRules[{Times, Plus, Subtract, divide, power, root, log}, 4];

and run legitCombos as before. The result? 595 hands are solvable, as opposed to the 566 we had before. hands that can be solved now, that couldn’t be solved before, include

{1,1,8,9} -> 8 Root[9,1+1] and trivial variations
{1,2,9,9} -> (9-1) Root[9,2] and trivial variations
{2, 2, 2, 6} -> 2^(2+Log[2,6]) or (2*Root[6,2])^2 and trivial variations
{2,2,9,9} -> 2(Root[9,2]+9) and trivial variations
{3,7,8,10} -> 7 Root[8,3] +10 and trivial variations
{3,4,9,10} -> 4+Log[3,9]*10 and trivial variations
{7, 8, 9, 9} -> 8 Root[9,9-7] and trivial variations, and
{8, 8, 9, 10} -> 8 Root[9,10-8] and trivial variations.

The hands that remain unsolvable are

{{1, 1, 1, 1}, {1, 1, 1, 2}, {1, 1, 1, 3}, {1, 1, 1, 4}, {1, 1, 1, 6}, {1, 1, 1, 7}, {1, 1, 1, 9}, {1, 1, 1, 10}, {1, 1, 2, 2}, {1, 1, 2, 3}, {1, 1, 5, 9}, {1, 1, 5, 10}, {1, 1, 6, 7}, {1, 1, 6, 10}, {1, 1, 7, 7}, {1, 1, 7, 8}, {1, 1, 7, 9}, {1, 1, 8, 10}, {1, 1, 9, 9}, {1, 1, 9, 10}, {1, 1, 10, 10}, {1, 2, 2, 2}, {1, 2, 9, 10}, {1, 2, 10, 10}, {1, 4, 9, 9}, {1, 5, 7, 7}, {1, 6, 6, 7}, {1, 6, 7, 7}, {1, 6, 7, 8}, {1, 6, 10, 10}, {1, 7, 7, 7}, {1, 7, 7, 8}, {1, 7, 10, 10}, {1, 8, 9, 9}, {1, 8, 9, 10}, {1, 8, 10, 10}, {1, 9, 9, 9}, {1, 9, 9, 10}, {1, 9, 10, 10}, {1, 10, 10, 10}, {2, 2, 2, 2}, {2, 2, 7, 9}, {2, 6, 7, 7}, {2, 7, 7, 7}, {2, 7, 7, 9}, {2, 7, 8, 10}, {2, 7, 9, 9}, {2, 9, 9, 9}, {2, 9, 9, 10}, {2, 10, 10, 10}, {3, 3, 4, 10}, {3, 3, 10, 10}, {3, 4, 6, 7}, {3, 5, 5, 10}, {3, 5, 7, 7}, {3, 5, 8, 10}, {3, 10, 10, 10}, {4, 4, 5, 9}, {4, 4, 6, 6}, {4, 4, 6, 7}, {4, 4, 9, 9}, {4, 4, 9, 10}, {4, 7, 7, 9}, {4, 7, 7, 10}, {4, 9, 9, 9}, {4, 9, 10, 10}, {4, 10, 10, 10}, {5, 5, 5, 7}, {5, 5, 5, 8}, {5, 5, 5, 10}, {5, 5, 6, 9}, {5, 5, 6, 10}, {5, 5, 7, 9}, {5, 6, 7, 10}, {5, 7, 7, 7}, {5, 7, 7, 8}, {5, 7, 9, 9}, {5, 8, 9, 9}, {5, 8, 10, 10}, {5, 9, 9, 9}, {5, 9, 9, 10}, {5, 10, 10, 10}, {6, 6, 6, 7}, {6, 6, 7, 7}, {6, 6, 7, 8}, {6, 6, 10, 10}, {6, 7, 7, 7}, {6, 7, 7, 8}, {6, 7, 7, 9}, {6, 7, 8, 8}, {6, 8, 10, 10}, {6, 9, 9, 9}, {6, 9, 10, 10}, {7, 7, 7, 7}, {7, 7, 7, 8}, {7, 7, 7, 9}, {7, 7, 7, 10}, {7, 7, 8, 8}, {7, 7, 8, 9}, {7, 7, 8, 10}, {7, 7, 9, 9}, {7, 7, 10, 10}, {7, 8, 8, 8}, {7, 9, 9, 9}, {7, 9, 9, 10}, {7, 9, 10, 10}, {7, 10, 10, 10}, {8, 8, 8, 8}, {8, 8, 8, 9}, {8, 8, 9, 9}, {8, 8, 10, 10}, {8, 9, 9, 9}, {8, 9, 9, 10}, {8, 9, 10, 10}, {8, 10, 10, 10}, {9, 9, 9, 9}, {9, 9, 9, 10}, {9, 9, 10, 10}, {9, 10, 10, 10}, {10, 10, 10, 10}}.

Next chapter, we add Mod[].