Posts Tagged ‘monomatching’

Monomatch Dice

July 12th, 2021

A game called “Spot It!” has received a lot of attention from recreational mathematicians in recent years. There’s a good video by Matt Parker, and a blog post about it by one of my five readers. (Hi MJD!) The game contains a number of cards each with eight symbols, and the object is to be the first to spot the matching symbol between the pair of cards. The designers of the game, in finding an elegant set of cards where every pair has exactly one match, used a structure that can be understood in terms of finite projective geometry. Parker calls these monomatch sets of symbol sets.

This led me to consider monomatching on symbols on the faces of a pair of dice. (I guess I’m on a roll with the dice content here. Sorry, there’s really only one decent dice related pun; if you try to stretch beyond that, things get dicey.) The obvious thing to do with a pair of d6’s is to have a set of 36 symbols. You could consider the symbols as being arranged in a 6×6 grid. The faces on one die would each contain all of the symbols on one row, and the faces on the other die would each contain the symbols on one column. For an added trick, the numbers 1 through 6 could be six of the symbols, and if they are on a diagonal of that grid, the dice could be used as regular dice by ignoring the non-numerical symbols.

This seems like an idea worth exploring, once my laser engraver arrives. There is a version of Spot It! with 30 cards and 6 symbols per card that is marketed to be played by young children, so it seems like it would be somewhere near the realm of playability. (Adding dice beyond the first two might help.) And although Spot It! already comes in a compact tin, there’s not much that’s more portable than a pair of dice. But there’s not really any kind of interesting puzzle to be found in it, so I was hoping to find something else to do with symbol matching on dice.

And I did come up with another idea, and it is good, and it is dumb. Imagine, if you will, that you could use a pair of dice as… 2d6!

Here, unlike with Spot It!, it’s desirable to make spotting the match be easy, so I’ve put the numbers in order and used colors as an aid.

But not, of course, 2d6 as we know it. Instead of adding numbers on the two dice, we’d have the numbers 2 through 12 as symbols for matching. The frequencies in which the numbers occur on the two dice would have to be such that the probability of getting a number as a match would be the same as the probability of getting that number as a sum using a regular pair of d6’s.

2d6 rollFrequencyFaces per die
211, 1
321, 2
431, 3
542, 2 or 1, 4
651, 5
762, 3 or 1, 6
851, 5
942, 2 or 1, 4
1031, 3
1121, 2
1211, 1

Now, finding a set of number sets for the symbols on the faces of each die becomes an interesting puzzle, especially if we add constraints to make our dice more nice. One type of constraint we might care about is on the quantity of numbers on each die. Minimizing the total quantity on both of the dice would be good, as would be balancing the quantities on the dice.

Unfortunately, we cannot do both. The minimum total quantity is 43, which is odd. So in order to balance the quantities, we need to use the inefficient alternative for either 5 or 9. (Using the inefficient alternative for 7 doesn’t change the parity of the total, so I’ve dismissed that option.)

We could try to go further in our pursuit of balance. The solution I found above has balanced number quantities on the two dice, but at the level of faces, there are issues. The first set has face quantities of {2, 3, 3, 4, 5, 5}, while the second has {2, 3, 4, 4, 4, 5}. Having the same face quantities between the two dice would be desirable, especially if it could be done while minimizing the number of faces with five numbers, since those faces look more cluttered.

You could also drill down to the numbers themselves. The sum of all of the numbers on the upper die above is 158, while the sum on the lower die is 148. Ideally, we’d make those sums equal, or at least closer.

Problem #50: Find a “nicer” numbering for a pair of monomatch 2d6 dice than the one I found. One potential flaw that I haven’t mentioned already is having a pair of faces on the same die with identical number sets. When I was manually looking for numberings, they seem to want to have pairs like this, so they are harder to avoid than you might think.