Optimal Bidding Strategy for Casino Card Game

This brain teaser describes a casino card game where you and the house accumulate cards into separate piles based on their colors, with mixed-color pairs removed from play. The twist is that you are allowed to choose the entry fee yourself, but you only get a fixed payout if you end up with strictly more cards than the dealer. Candidates must reason about the overall structure of the game rather than simulate individual rounds, and recognize that the randomness comes only from the order in which cards appear, not from the deck's overall composition.

Solving it relies on identifying and exploiting an invariant: a quantity that the game's rules cannot change, even though they reshuffle where cards end up. Strong answers connect the color counts in the original deck to the final possible configurations of the two piles and discarded cards. Interviewers look for the ability to formalize intuitive symmetry arguments, avoid being misled by path-dependent thinking, and convert qualitative invariance into a clear statement about expected value and thus an optimal maximum price to pay.

What it tests

This class of problems is governed by the principle of invariants: certain quantities remain unchanged despite the operations performed. Here, the key invariant is the difference between the number of red and black cards, which is fixed by the deck's composition. The rules for moving cards to piles or discarding them are constructed so that every action either removes equal numbers of red and black cards (in the case of discards) or segregates them into separate piles without altering their total counts. Because the initial deck is perfectly balanced, and every operation preserves this balance, the final outcome is constrained by this invariant, regardless of the sequence of pairs revealed.