The blind man and the cups

[clementine]

A blind man and a fool decide to play a game. Four cups are placed at the corners of a square table. The blind man gets to choose a subset of the four cups and flip them simultaneously. Effectively, he may choose “one cup”, “two diagonally opposite cups”, “two adjacent cups”, “three cups” or “four cups” to flip. If, after flipping, all four cups are upright, the blind man wins the game! Otherwise, the game continues: the fool rotates the table by an amount of his choice. Can the blind man win the game? Present your plan below.

[Blazer]

im just gonna say NO HE"S BLIND! xD

[Auckmid]

Flip all 4 cups?

[AppleyNO]

Can the blind man touch the cups beforehand?

[AppleyNO]

Flip all 4 cups?

Thats assuming that all four cups are facedown

[Auckmid]

Thats assuming that all four cups are facedown

Hmm, wasn't thinking about that...

[Glip]

This is actually pretty easy for the blind man, if he can tell by touch when he flips a cup if it is facing up or down.


First, you need to flip all four cups to find out the arrangement of the cups by touch. There are essentially only 4 different starting positions: No odd cups, 1 odd cup, 2 adjacent odd cups, or 2 diagonal odd cups. I'll show a binary representation of all variants, with 0 representing cups facing one way and 1 representing cups facing the other.

Variant 1 - No Odd Cups

00
00

Flip all four and win!

Variant 2 - One Odd Cup

00
01

Step 1: Flip Three. You either win, or are in one of the following arrangements. You know which arrangement based on whether the odd cup (detected by touch) was one of the first or last cups you flipped (Arrangement A), or if it was the middle one (Arrangement B), or none of them (Arrangement C).

A. B. C.
10 10 00
10 01 00

Step 2-A: Flip Two Adjacent. Again you'll know arrangement for next step if no win. Step 2-B: Flip Two Diagonal, then Flip All Four if you didn't already win! Step 2-C: Flip all four and win!

A1 A2
00 10
00 01

Step 3-A1: Flip all four and win! Step 3-A2: See Step 2-B.

Variant 3 - Two Adjacent Odd Cups

10
10

See Steps 2-A and following.

Variant 4 - Two Diagonal Odd Cups

10
01

See Step 2-B.



If he can't tell by touch if the cups are facing up or down, that's a different story, but still doable.




Variant α - Unknown

A. B. C. D.
00 00 10 10
00 01 10 01

Step 1: Flip All Four twice. This will win in Starting Position A without changing the arrangement of B, C, or D.

B. C. D.
00 10 10
01 10 01

Step 2: Flip Two Diagonal, then Flip All Four. This will win Starting Position D, without actually changing B or C. (Try it yourself. B ends up with 1 odd and C ends up with two adjacent odd, regardless of which two diagonals are flipped.)

B. C.
00 10
01 10

Step 3: Flip Two Adjacent, then Flip All Four. This will either win C or leave C in the following arrangement, without changing B.

B. C.
00 10
01 01

Step 4: Flip Two Diagonal, then Flip All Four. This will win C without changing B.

B.
00
01

Step 5: Flip Three, then Flip All Four. This will win B or leave it in one of the following arrangements.

B1 B2
10 10
10 01

Step 6: Flip Two Diagonal, then Flip All Four. This will win if it ended up in B2, without changing it if it ended up in B1.

B1
10
10

Step 7: Flip Two Adjacent, then Flip All Four. This will either win, or leave it in the following arrangement.

B1
10
01

Step 8: Flip Two Diagonal, then Flip All Four. You win!

[clementine]

Can the blind man touch the cups beforehand?

I think that's another variant of this puzzle.

Also Glip solved it! Well done! You are quite the puzzlemaster.

[Glip]

I just thought of a new variant for this in which the blind man not only can't tell whether a cup is facing up or down by touching it, but he won't be told when he has won, and instead must announce it for himself, or lose. However, as a saving grace, he doesn't have to make sure all cups are facing upward, just that all cups are facing the same direction.

I'm going to ponder upon this for awhile and see if it's possible.

SAD EDIT: Just finished thinking it through. It's not possible :*(