Can You Solve The Cursed Dice Riddle?

Introduction

Spring, the favorite season of Demeter, the goddess of the harvest, is a time when humans and animals look to her to balance the bounty of the natural world. As any self-respecting goddess would, she accomplishes this task with a pair of magical dice. Every day at dawn, she rolls the dice, and all lands that match the sum of the two dice produce their resources. The resulting frequency of sums across the season keeps her land in perfect harmony; any other rates would spell ruin.

The Trickster God’s Curse

However, this harmony was disrupted when Loki, the Norse trickster god, invaded Demeter’s land and cursed her dice, causing all the dots to fall off. When Demeter tried to reaffix them, she found that one die wouldn’t accept more than four dots on any of its sides, though the other had no such constraint. She could use Hephaestus’ furnace to seal the dots in place before the sun rises, but once sealed, she couldn’t move or remove them again. The challenge was to craft the dice so that, when rolled and summed, every total comes up with the exact same frequency as it would with standard 6-sided dice.

The Solution

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. By making a table with all the possibilities for one die represented on the top and those for the other on the side, we can see the odds of rolling any sum. This approach helps in crafting a new set of dice. The locations of the sums in this table may change, but the numbers and quantities of each sum must stay the same.

To start, each die needs a 1 on it to roll a 2. 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. 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.

Finalizing the Dice

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 Sicherman 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. With the dice reforged, Demeter could now be confident that she had averted disaster. Now it was time to repay the Norse gods with a gift of her own.