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Chapter 2
Sets, Functions, and Reasoning
MATH ACTIVITY 2.3
Deductive Reasoning Game
Pica-Centro is a game for two players (or groups) in which one player uses deductive reasoning (obtaining conclusions from given information) to determine the digits of a number
that is selected by the other player.* The game begins with player A choosing a three-digit
number that contains no zeros and recording it on a slip of paper without showing it to
player B. Player B then tries to determine this number by asking questions. Player B
records all guesses and player A’s responses in the table. Player B’s first attempt is a guess,
but after this, deductive reasoning is used based on Player A’s replies.
NCTM Standards
Beginning in the elementary
grades, children can learn to
disprove conjectures by finding
counterexamples. At all levels,
children will learn to reason
inductively from patterns and
specific cases. Increasingly over
the grades, they should also learn
to make effective deductive
arguments. p. 59
*1. Suppose player A chooses 574 and player B’s first guess is 123 (see table below). Then
player A responds by saying 0 pica and 0 centro, and player B records this response in
the first row of the table. A pica is a digit that is correct but not in the correct position,
and a centro is a digit that is both correct and in the correct position. What does player
B know from this response?
*2. Player B’s second guess is 456 (see table) and player A responds with 2 pica and 0 centro, which player B records in the second row of the table. What does player B know
from this response?
*3. A logical third guess for player B is 654 or 546 or 465. Explain why.
*4. Suppose player B chooses 654 (see table). Then player A would say 1 pica and 1 centro. What can player B deduce from this information? Explain your reasoning. Can
player B conclude that the digit 4 is in the correct position? Can player B conclude that
5 is one of the digits in player A’s number?
Guesses
Digits
Responses
Pica
Centro
1
2
3
0
0
4
5
6
2
0
6
5
4
1
1
5. Make a Pica-Centro table and play this game with another person. Then reverse roles
so that both players have a chance to use deductive reasoning. The player who requires
the fewest guesses to determine the number is the winner.
*D. B. Aichele, “Pica-Centro, A Game of Logic,” The Arithmetic Teacher 19, no. 5 (May 1972): 359–361.