This is the eighteenth puzzle in Matt Parker's Matt Parker's Maths Puzzles puzzle series
A contestant is on a game show with five different prizes:
- A car;
- A diamond;
- A goat;
- A jar of bogeys; and,
- A poke in the eye.
The contestant can ask for any prize that they want, except that when they ask for a prize, they will be given two other prizes (that are not what they asked for). The contestant can ask for any number of prizes, but if they are to be given the same prize twice, it will cancel, and they will not recieve it at all.
To help the contestant out, the host also gives the following information:
- If you ask for the car and the goat; you will win bogeys and a poke in the eye.
- If you ask for the car, bogeys and a poke in the eye; you will win the car, bogeys, a poke in the eye, and the diamond as well!
- Two of the things will give the diamond as a prize.
- Two of the things will give the car as a prize.
- For all prizes, there are exactly two items that, when asked for, will yield that prize.
What should the contestant ask for in order to win only the car and the diamond?
and
How few things can the contestant ask for in order to win them?
We can make a table of the two items that will yield each prize.
| Prize | Item 1 | Item 2 |
|---|---|---|
| car | ? | ? |
| diamond | ? | ? |
| goat | ? | ? |
| bogeys | ? | ? |
| eye-poke | ? | ? |
From the information that the host gave us:
If you ask for the car and the goat; you will win bogeys and a poke in the eye.
We know that:
- Exactly one of the car and goat yield bogeys.
- Exactly one of the car and goat yield a poke in the eye.
- Since no object can yield itself, both the car and the goat must yield the diamond.
We can now update our table:
| Prize | Item 1 | Item 2 |
|---|---|---|
| car | ? | ? |
| diamond | car | goat |
| goat | ? | ? |
| bogeys | car or goat | ? |
| eye-poke | goat or car | ? |
The host also told us:
If you ask for the car, bogeys and a poke in the eye; you will win the car, bogeys, a poke in the eye, and the diamond as well!
Asking for the car gives the diamond. Asking for the car will yield two prizes, though, and the first piece of information lets us know that the second prize must be the bogeys or the poke in the eye.
Suppose the car yields the bogeys. Then the poke in the eye cannot also yield the bogeys, as otherwise the bogeys would not be a prize. It cannot yield the diamond as otherwise there would be more that two things that yield the diamond. And it cannot yield itself. This means that the poke in the eye must yield the car and the goat.
Also, since the poke in the eye yields the car, and the car is a prize, the bogeys cannot yield the car. The bogeys cannot yield the diamond either, and the bogeys cannot yield themselves. The two remaining spaces would be yielded by the diamond.
This would give us the following table:
| Prize | Item 1 | Item 2 |
|---|---|---|
| car | eye-poke | diamond |
| diamond | car | goat |
| goat | eye-poke | bogeys |
| bogeys | car | diamond |
| eye-poke | goat | bogeys |
So we can win the car and the diamond by asking for:
- The car (giving us the diamond and the bogeys); and,
- The diamond (giving us the car and the bogeys).
We won both the car and the diamond once, and the bogeys twice, so we will recieve the car and the diamond only.
Suppose instead that the car yields the poke in the eye. The bogeys thus cannot yield the poke in the eye, as it is a prize. They cannot yield themselves. And they cannot yield the diamond. So they must yield the goat and the car.
Since the car is a prize, the poke in the eye cannot yield the car. It cannot yield the diamond. And it cannot yield itself. Thus it must yield the goat and the bogeys.
That would give us this table:
| Prize | Item 1 | Item 2 |
|---|---|---|
| car | bogeys | diamond |
| diamond | car | goat |
| goat | bogeys | eye-poke |
| bogeys | goat | eye-poke |
| eye-poke | car | diamond |
We can win the car and the diamond by asking for:
- The car (giving us the diamond and the poke in the eye); and,
- The diamond (giving us the car and the poke in the eye).
We won both the car and the diamond once, and the poke in the eye twice, so we will recieve the car and the diamond only.
In both cases, we won by asking for the same things, so it doesn't matter which case occurs in real life. If we ask for the diamond and the car, we will win the diamond and the car.
If we could win them in less that two asks, we must win them in zero or one asks. We get nothing from zero asks. To win them in one ask there must be some item that yields them both. Since there is no such item, the minimum number of asks that we can win the diamond and the car in is two.