Math Dice ships with the wrong rules.
Roll two twelve-sided dice and multiply them — that’s your target. Now hit it using three six-sided dice, each exactly once. I computed every possible game, and the box is selling you a worse version than the one you can play.
The finding
The published rules make it unsolvable two times in three
ThinkFun’s rules allow + − × ÷ and exponents. Played that way, an exact answer exists only 30.7% of the time. Most rounds, nobody can win properly and you settle for closest.
Allow two more operations that need no extra dice — factorial and square root — and the same game becomes solvable 64.4% of the time. Two symbols, and the game more than doubles.
Every round is a negotiation about who got closest.
Most rounds now have a right answer, and finding it is the game.
Those percentages are weighted by how often each roll and each target actually come up. Counting all 4,368 distinct rolls equally gives 60.7% — lower, which tells you the rolls you see most often are the kinder ones.
Things that fell out of it
Four results I didn’t expect
The hardest roll in the game
6 × 11 = 66, with 4 4 4. The only route is √(√(√4^4!)) + √4 — make a 2, raise it to 4! = 24 to reach 16,777,216, then take three square roots back down. Two rolls out of 4,368 are this bad.
Seven numbers no roll can reach
89, 103, 106, 107, 133, 134 and 137 are unreachable from any three dice. Not one of them is a possible d12 × d12 product, so the game never asks. That is luck, not design.
Your dice decide the round
3 4 5 reaches 54 of the 59 possible targets. 1 1 1 reaches four. Before anyone has thought about anything, the round is largely settled.
Cube roots are worthless
Adding n-th roots to the rules produces zero new answers anywhere. A satisfying result: sometimes the honest finding is that your idea does nothing.
The answers
Every roll, every target, one worked solution
Exhaustive search over every expression, in exact fractions rather than floating point, with the limits raised until the answer stopped changing. Then all 39,135 printed expressions were re-checked by a second parser written independently of the solver — each one has to equal the number it’s filed under and use each die exactly once. Zero failures.
The ten hardest rolls
A challenge sheet, answers on the back. Every one is exactly solvable, which is what makes them cruel. PDF, 1 page
The solution book
One simplest answer for every roll against every target, plus the nearest miss where there’s no exact hit. PDF, 5 pages
With the house rules
The same book allowing concatenation and decimals — two dice reading as 35, or as 3.5. PDF, 6 pages
What “simplest” means
Not my opinion. I generated the contested cases, picked the ones that felt easiest, and fitted the ranking to those answers. Fewest operators wins, and steps that can’t change the number are free.
Play it
You need five dice and nothing else
Two d12 and three d6. Multiply the twelves, hit it with the sixes, and allow factorial and square root or you’re playing the bad version. Closest wins when nobody can do it exactly.
No dice to hand? There’s a browser version — a puzzle a day, three rule sets, and it checks your answer in exact fractions rather than decimals, so it can’t be fooled by a rounding error.
I will absolutely play the real thing with you. Email me — or if you’d rather do something more strenuous first, there’s running and lifting.