Keep That Card in Mind: Card Guessing with Limited Memory

Boaz Menuhin, Moni Naor

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


A card guessing game is played between two players, Guesser and Dealer. At the beginning of the game, the Dealer holds a deck of n cards (labeled 1,..., n). For n turns, the Dealer draws a card from the deck, the Guesser guesses which card was drawn, and then the card is discarded from the deck. The Guesser receives a point for each correctly guessed card. With perfect memory, a Guesser can keep track of all cards that were played so far and pick at random a card that has not appeared so far, yielding in expectation ln n correct guesses, regardless of how the Dealer arranges the deck. With no memory, the best a Guesser can do will result in a single guess in expectation. We consider the case of a memory bounded Guesser that has m < n memory bits. We show that the performance of such a memory bounded Guesser depends much on the behavior of the Dealer. In more detail, we show that there is a gap between the static case, where the Dealer draws cards from a properly shuffled deck or a prearranged one, and the adaptive case, where the Dealer draws cards thoughtfully, in an adversarial manner. Specifically: 1. We show a Guesser with O(log2 n) memory bits that scores a near optimal result against any static Dealer. 2. We show that no Guesser with m bits of memory can score better than O(√m) correct guesses against a random Dealer, thus, no Guesser can score better than min{√m, ln n}, i.e., the above Guesser is optimal. 3. We show an efficient adaptive Dealer against which no Guesser with m memory bits can make more than ln m + 2 ln log n + O(1) correct guesses in expectation. These results are (almost) tight, and we prove them using compression arguments that harness the guessing strategy for encoding.

Original languageEnglish
Title of host publication13th Innovations in Theoretical Computer Science Conference, ITCS 2022
EditorsMark Braverman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Number of pages28
ISBN (Electronic)9783959772174
StatePublished - 2022
Event13th Innovations in Theoretical Computer Science Conference, ITCS 2022 - Berkeley, United States
Duration: 31 Jan 20223 Feb 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference13th Innovations in Theoretical Computer Science Conference, ITCS 2022
Country/TerritoryUnited States

All Science Journal Classification (ASJC) codes

  • Software


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