The Enigna of the Cursed Dice

Spring has arrived, and for Demeter, the goddess of the harvest, it's a season of utmost significance. The responsibility falls upon her shoulders to maintain equilibrium in the natural world's abundance, a task she accomplishes with a pair of mystical dice. Each morning, as the sun breaks over the horizon, Demeter casts these dice, and the lands corresponding to the sum of their faces yield their resources. This harmony in the distribution of sums is crucial, for any deviation from the norm could lead to catastrophe.

However, the tranquility of this divine ritual was shattered when Loki, the mischievous Norse god, invaded Demeter's domain and placed a curse upon her dice. As a result, all the dots on the dice vanished. In her attempts to restore them, Demeter discovered that one of the dice could only bear up to four dots on any given side, while the other was unaffected by such constraints. She could use Hephaestus' furnace to permanently seal the dots in place before daybreak, but once sealed, there would be no altering or removing them.

The predicament was clear: how could Demeter craft her dice in such a way that, when rolled and summed, every possible total would occur with the exact same frequency as it did with standard 6-sided dice?

To solve this conundrum, let's delve into the mechanics. Regular dice can yield sums ranging from 2 to 12, with intermediate sums occurring more frequently than those on the extremes. By constructing a table that lists all the possible outcomes for one die on the top and another on the side, we can visualize the odds associated with each sum. For instance, there are six ways to roll a sum of 7, but only two ways to achieve a sum of 3.

To create the new set of dice while maintaining the same sum frequencies, we must ensure that the sums' positions in the table may change, but the numbers and quantities of each sum must remain constant. In simpler terms, there should still be precisely one way to achieve a sum of 2, two ways to attain a sum of 3, and so forth.

The first step is to secure a sum of 2, and since only positive whole numbers are allowed, each die must possess a side with a single dot. We can deduce that, on the cursed die, the highest possible number is 4, while the other die has no such restriction. Consequently, one die must bear a 1 and a 4 to ensure a single way to roll a 12, avoiding overrepresentation of sums like 2 or 12.

With this knowledge, we deduce that the remaining four sides of the cursed die must contain a combination of 2s and 3s. If we have three or four 2s, we'll roll a sum of 3 too frequently. Likewise, if we have three or four 3s, a sum of 11 will be too common. Hence, the only viable configuration for the cursed die is two 2s and two 3s.

Now, with one die's arrangement complete, we can ascertain the missing values for the second die. To achieve one more way to roll a 10 and a 4, we must include one 3 and one 6. The final requirement is to secure one more way to roll a 5 and a 9, which compels us to employ the numbers 4 and 5 for the remaining sides.

When we finalize this arrangement, we find ourselves with a distribution table in which every possible sum appears with the same frequency as with standard dice. This remarkable discovery was credited to George Sicherman in 1978, leading to their designation as "Sicherman dice." Astonishingly, this remains the sole alternative configuration to replicate the distribution of sums found in traditional 6-sided dice.

With the newly forged dice in hand, Demeter is now confident that she has averted disaster. It's time for her to express her gratitude to the Norse gods with a gift of her own.