Can you solve the cursed dice riddle? - Dan Finkel

Ah, spring. As Demeter, goddess of the harvest, it’s your favorite season. Humans and animals look to you to balance the bounty of the natural world, which, like any self-respecting goddess, you do with a pair of magical dice. Every day you roll the dice at dawn, and all lands that match the sum of the two dice produce their resources. The resulting frequency of sums across the season keeps your land in perfect harmony; any other rates would spell ruin.

And that’s why it was particularly rotten when Loki, the Norse trickster god, invaded your land and cursed your dice, causing all the dots to fall off. When you try to reaffix them, you find that one die won’t accept more than four dots on any of its sides, though the other has no such constraint. You can use Hephaestus’ furnace to seal the dots in place before the sun rises, but once sealed you can’t move or remove them again.

How can you craft your dice so that, when rolled and summed, every total comes up with the exact same frequency as it would with standard 6-sided dice? Pause here to figure it out for yourself.

Answer in 3. Answer in 2. Answer in 1. Normal dice can roll any sum from 2 to 12, but sums in the middle tend to come up more frequently than ones on the ends. We can see the odds of rolling any sum by making a table, with all the possibilities for one die represented on the top, and those for the other on the side. The table lets us see at a glance that there are six ways to roll a 7, but only two ways to roll a 3.

This also gives us an approach to crafting our new set of dice. Matching the original sum frequencies means that the locations of the sums in this table may change, but the numbers and quantities of each sum must stay the same. In other words, there still must be exactly one 2, two 3s, and so on.

To start, we’ve got to roll that 2, and since we have to use positive, whole numbers, there’s only one choice: each die needs a 1 on it. What else do we know? Assuming we have a 4—the highest number possible—on the cursed die, the other one would need an 8 in order to have one way to roll 12.

In fact, we know we require a single 1 and a single 4 on the cursed die, or we’d have too many ways to roll a 2 or a 12. So the cursed die's remaining four sides must have a mix of 2s and 3s. If we have three or four 2s, we can roll the sum 3 too many ways. Similarly, if we have three or four 3s, we’d get the sum 11 too often.

So the only possibility is for the cursed die to have two 2s and two 3s. With one die completed, we should be able to figure out the missing values on the second. First, we need one more way to make 10 and 4, so we must have one 3 and one 6. We now need one more way to make 5 and 9. That forces us to choose 4 and 5 for the final sides.

Fill them in, and lo and behold, we have a distribution table where every possible sum shows up the same number of times as with our original dice! The discovery of these dice was made in 1978 by George Sicherman, which is why they’re referred to as “Sicherman dice.” Incredibly, this is the only alternate way to make two 6-sided dice with the same distribution of sums as standard dice.

You send the dice to be reforged, confident that you’ve averted disaster. Now it’s time to repay the Norse gods with a gift of your own.