Problem:
Seven students count from to as follows:
Alice says all of the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
Debbie, Eliza, and Fatima say all of the numbers that none of the students with first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
Finally, George says the only number that no one else says.
What number does George say?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
After each person counts, the numbers left for the next person form an arithmetic progression. For example, Alice leaves all of the numbers for Barbara. If a student leaves the progression , then the next student leaves the progression
This implies that in the following table, each number in the third column is three times the previous entry in the third column, and each entry in the second column is the sum of the two entries in the row above:
George is left with the single term .
The numbers skipped by Alice are the middle numbers in each consecutive group of that is, and so on. The numbers skipped by Alice and Barbara are the middle numbers in each group of , that is, , and so on. In general, the numbers skipped by all of the first students are the middle numbers in each group of . Because , the only number not exceeding that is skipped by the first six students is . That is the number that George says.
The problems on this page are the property of the MAA's American Mathematics Competitions