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 dies 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.
