You
nerdsniped me.
NOTE: my answer is obfuscated and inefficient. feel free to make it more efficient so that everyone can learn new things. I wanted to try and figure out a way to narrow down which pairs you'd actually have to guess for the first pair, so if you guys ever get a similar problem but dealing with 4000-sided dice, this approach might be helpful.
Okay so what we know so far in this problem is that there's two dies, X and Y.
your score is basically just the product of whatever you roll for X & Y in a round. So for the first particular game it mentions, there's only a few ways that one configuration of dice rolls can be exactly 5 points higher than another.
If you list out all the possible products of dice rolls, this is what you come up with:
36
30
25
24
20
18
16
15
12
10
9
8
6
5
4
3
2
1
Now I can highlight pairs of numbers with a difference of 5 in products
36
30
25
24
20
18
16
15
12
10
9
8
6
5
4
3
2
1
Now, since dice rolls have to be whole number integers, my next intuition is to come up with numbers that can only create these products of pairs. We can ignore order in our permutations since the problem doesn't ask about specific dice. As you can see, the numbers I've grouped in yellow are actually ones that can only be produced (by dice) if one of the dies is '5'.
Additionally, for the orange pair, '3' can only be created as a product if one of the dies is 1 and the other is 3, and '8' can only be produced if you roll a 2 and a 4.
Now, we've run out of pairs that come from nice numbers and we're left with stuffty, complicated ones with multiple answers.
With the green pair, '9' can only come from 3 and 3, and '4' can come from 2 & 2 or 1 & 4
With red, the '1' obviously comes from 1 and 1, but '6' can be from 1 & 6, or 2 & 3.
Now let's do algebra:
For the sake of this problem, let's just say that P
1,2...5 represent the products of each roll.
So as the problem says,
P
1 = P
2 -5
P
2 = P
3 + 6
P
3 = P
4 - 11
P
4 = P
5 + 8
Now here comes the cool part, you can rewrite all of these using the algebraic transitive property so that they all equal P
1. Then, you just solve, and whichever pair for P1 doesn't make another pair with prime numbers (which [besides 5] are impossible to roll with dice) or numbers that can't be made with products of 1 through 6, is the right answer.
Combinations thus far:
second roll/ first roll
5 & 6/5 & 5
5 & 5/5 & 4
5 & 4/5 & 3
5 & 3/5 & 2
5 & 2/5 & 1
2 & 4/3 & 1
3 & 3/2 & 2
3 & 3/1 & 4
1 & 6/ 1 & 1
2 & 3/1 & 1
Now, there's nothing really preventing us from plugging in the pairs of the second roll into the equations, but one of the possible pairs that was rolled in the second turn appears twice. So let's just use the first roll.
Now, I've rewritten all of the equations so that each subsequent turn is written in an equation with P
1 in it.
P
1 + 5 = P
2 P
1 - 1 = P
3P
1 + 10 = P
4P
1 + 2 = P
5Looking at this, it's obvious that the product of the first turn can't be greater than 26 or less than 1 because it would create other products that don't make sense for dice.
So when I plug in '25' for the first one, it tells me P
2 is 30, which is plausible.
The second gets me '24' for the third turn, which is also plausible
Yet the third gets me 35, which is bullstuff, so the first pair of numbers for roll 1 isn't 5 & 5.
Now finally, if you plug in the values for each pair that we came up with, you find that the only one that creates answers that don't have at least one prime number or something impossible like '14', you can deduce that the value of the first pair is 10. Then, you just take that and get the rest of the pairs.
ANSWER:
First roll = 10
Second roll = 15
Third roll = 9
Fourth roll = 20
Fifth roll = 12
