A brain teaser

riddles 28 December 2012

These two brain teasers posted on /r/math impressed me greatly, so I decided to present them here.

You have been abducted by terrorists and blindfolded. They sit you down at a table and tell you that if you can solve two puzzles then they will let you go.

There is a deck of 52 cards on the table in front of you. Ten of them are face-up and the rest are face-down. Split the deck such that each new stack has an equal number of face-up cards.

You sit thinking in darkness for a few minutes. The terrorists are growing impatient around you when you finally smile: you’ve got the solution! You count out 10 cards from the deck and flip them over. Then you push the two stacks across the table.

The terrorists count the face-up cards in each stack suspiciously. When they find that the number of face-up cards is identical, they’re impressed, but you have a yet harder challenge to face before they will let you go.

On one side of the table there are five dice which add up to 15. On the other side of the table there are another five dice which add up to 13. Move some dice around so that the two sides of the table have equal sums.

You furrow your brow and sit thinking in darkness for over an hour. The terrorists’ new respect for you is wearing off quickly, and they won’t wait much longer. Finally, you get it. You take one die from the group on the right and move it to the group on the left. Then you flip the remaining 4 dice of the right group upside-down. “You see,” you explain carefully. “If the left group sums to $k$, then the right group sums to $28-k$; this was given in the problem. If you take a die $b$ from the right group and move it to the left, then the new sums are $k+b$ and $28-k-b$. Now, because the opposite sides of a single die always add to 7, any time you flip a set of 4 dice summing to $x$, you end up with a set of 4 dice summing to $28-x$. Therefore, if we flip all of the dice in the right group, we now have:” $28-(28-k-b)=k+b$ “which is the same value that we have on the left side.”