Solvability check: can every target card actually be built?

Math Attack, by Formative Minds, is a digital card game aimed at primary school students. The player picks cards from a random draw of 5 white cards (numbers 2 to 12) to form an equation that equals one of 5 green target cards (drawn from 45 targets between 0 and 84). The objective is to build as many target cards as possible within the time limit of 5 minutes.

problem

In a timed card game with random draws, some green target cards could turn out to be impossible to build from the white cards in hand, which would make the game unplayable.

This is complicated by the difference in difficulty between levels: lower primary (P1 and P2) works with smaller numbers and fewer times tables, and needs a minimum of 2 white cards, while P3 to P6 need a minimum of 3.

Playtesting by feel doesn't prove the game works every time. We needed to check every possible hand.

solution

The simulation tool enumerates all 3,003 possible hands, so the results are exact and not just estimates, and outputs the percentage of hands that can build each green card under that level's rules.

Why 3,003: a hand is 5 cards drawn from 11 possible numbers, where the order doesn't matter and numbers can repeat, which gives exactly 3,003 different hands. Each one is weighted by how likely it is to be dealt from the 110-card deck.

A card passes the test and is included in a level's pile only if at least 95% of hands can build it.

Why 95% and not 100%: even easy targets like 25 and 27 miss on a few rare hands, so a 100% rule would throw out cards that play perfectly well, and a player stuck with one of those rare hands can simply redraw. At 95%, every target up to 30 clears the bar for P1 and P2, while P3 to P6 can build all 45 from almost every hand.