Four friends, Holly, Belle, Carol, and Nick, gather for May birthdays. Holly announces that she has a game before dinner. She hid gifts for each of her friends inside three separate boxes secured with padlocks. She challenges her friends to figure out the combination without consulting each other.
She provides the following information. All the padlocks have the same combination. The padlocks use 3 digits from 0 to 9. She also tells them that the sum of the three digits is equal to nine, and every digit is equal to or greater than the previous digit. Holly tells each of her friends one of the digits in the combination. She states, “I’ve given the first digit to Belle, the second digit to Carol, and the third digit to Nick.” The caveat is that the friends cannot share their numbers with each other or they will forfeit the gifts.
Then Holly gives her friends 30 minutes to open the padlocks while she watches and finishes dinner.
The three friends begin to think of the solution. One by one, they each try their hand at their padlock, but none of them opens the padlock. Seeing that no one has succeeded, suddenly Carol realizes she knows the answer, and successfully opens her box, revealing a new fitness tracker. Following this, Nick opens his padlock, revealing a new tablet; and Belle opens her box to find new pair of headphones.
Having watched this entire event unfold, can you determine the correct combination?
Solution
A finite number of possible solutions exist for this problem that can be listed and then crossed off based on the player’s given digit. Three simple solutions exist where any player would have the combination by knowing just one single digit. These are the following: (reminder: only solutions summing to 9 and following the X<=Y<=Z property are valid)
0 0 9 (Carol and Nick would realize the answer from their given digit)
0 1 8 (Nick would realize the answer if he was given 8)
3 3 3 (Belle or Nick would know the combination if given this digit)
After 30 seconds have passed, each realizes none of the others knows the combination instantly. At this point Carol realizes the solution to the problem, since she has a digit where she could eliminate another player’s list of possible quick solutions (i.e. everyone now knows Nick did not have a 9, 8, or 3 and Belle did not have a 3.)
If Carol’s given digit was 1 she would know the only two possibilities are the following: (0, 1, 8) and (1, 1, 7). Because no player found the answer after 30 seconds, Nick did not have an 8, meaning Carol knew the combination was (1, 1, 7). A person exhausting all possible combinations and removing the obvious combinations will see this is the only set where Carol could know the answer. Therefore since Carol knew the answer, any observer could determine the combination as (1, 1, 7). As an FYI, Nick would likely know the answer a little faster than Belle (as he was able to eliminate more possibilities).