The 7 person game-show riddle

The scenario:

You and six of your friends are going on a game show. When you get there, you’ll be placed in a green room and told the rules of the game so that you all can strategize. Once you all agree on a strategy, you’ll be taken on stage where you’ll each be put in your own respective isolation chambers. Once your whole team is in their respective chambers, you will each be assigned a number between 1 and 7 (inclusive and possibly repeating). You’ll all be shown everyone else’s numbers, but not your own. Your whole team wins if any one member can guess his or her own number correctly.

What you know:

  1. Each member of your team is assigned a number between 1 and 7 inclusive and possibly repeating.
  2. Each member gets only one guess.
  3. There’s no penalty for wrong guesses.
  4. There’s absolutely no communication between members while the game is on.
  5. You all win if at least one person from your team guesses correctly.

Can you come up with a winning strategy?

Hints (click to unblur)

  1. Start with a simpler case (like only 2 players, then 3 players, etc.)

  2. Consider what is conserved.

  3. Consider modular arithmetic

A solution

A description

Regardless of who’s looking at the numbers, the sum of all the numbers (including your own) must be the same. The problem is that you don’t know what the missing number is (your own). But, if we make an assumption as to what the sum is, we could then solve for our own. But that leaves too many options (we could assume that the sum would be 7 or 49 or anything between). Luckily we don’t need all of that information. We really only need to assume something about what the sum is mod 7. For instance, if you knew that the sum of all the numbers was \(3 \pmod 7\), and you see the numbers 1, 2, 3, 4, 5, 6, (the sum of which is \(21 \equiv 0 \pmod 7\)), you would know that your number is 3. But that assumes that you know what the sum is (mod 7). Luckily you have a total of 7 people on your team!

The strategy

In the green room you assign a number from 1 to 7 (inclusive and not repeating) to each person in your team. So each person has a number and all numbers from 1 to 7 are represented. Then you all go into your respective isolation chambers and get your numbers and solve for your own assuming the sum mod 7 is whatever number you were assigned. Since the sum must be something from 1 to 7 mod 7, we know one person had the correct assumption, and since given the assumption, the solution is unique, we know one person will guess his or her own number correctly. And we’re done!