MA 490-505 Spring 2014
Recreational Problems
Problem 1 There are two egg timers: one for 7 minutes and one for 11 minutes. We must
boil an egg exactly 15 minutes. How can we do that using only these timers?
Problem 2 Distribute 127 one dollar bills among 7 wallets so that any integer sum from 1
through 127 dollars can be paid without opening the wallets.
Problem 3 Form a magic square with the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9; that is, place
them in the boxes of a 3 × 3 table so that all the sums of the numbers along the rows, columns,
and two diagonals are equal.
Problem 4 Using the digits 1, 9, 8, and 4 in the given order, and four arithmetic operations,
write numbers 2, 3, . . . , 8. Example: 5 = (1 + 9)/(8/4).
Problem 5 Write the number 2001 using only 4’s. Try to use as few digits as possible. You
may use as many arithmetic operations as necessary.
Problem 6 Write the number 100 using all the digits 1, 2, . . . , 9 only once. You may use as
many arithmetic operations as necessary.
Problem 7 Arrange parentheses in the expression
4 × 12 + 18 : 6 + 3
so that the value of the resulting expression is 50.
Problem 8 Twenty 5’s are written in a row. Can the signs + be placed between them, so that
the value of the resulting expression is 1000?
Problem 9 In an arithmetic addition problem the digits are replaced with letters (equal digits
by same letters, and different digits by different letters). The result is: LOVES+LIVE=THERE.
How many “loves” are “there” if I=2? The answer is the maximum possible value of the word
THERE.
Problem 10 Although the addition given below might appear valid, show that, in fact, there
is no substitution of distinct digits for the various letters which will give a numerically correct
statement:
T H R E E
+
F I V E
E I G H T
Problem 11 3 rings having different outer diameters are slipped onto an upright peg, the
largest ring on the bottom, to form a pyramid. We wish to transfer all the rings, one at a time,
to a second peg, but we have a third (auxiliary) peg at our disposal. During the transfers it is
not permitted to place a larger ring on a smaller one. Complete the transfer to peg number 2
using 7 moves.
Problem 12 4 rings having different outer diameters are slipped onto an upright peg, the
largest ring on the bottom, to form a pyramid. We wish to transfer all the rings, one at a time,
to a second peg, but we have a third (auxiliary) peg at our disposal. During the transfers it is
not permitted to place a larger ring on a smaller one. Complete the transfer to peg number 2
using 15 moves.
Topics are taken from Problem-Solving Strategies by Arthur Engel, 2007; and Mathematical
Circles: Russian Experience by Dmitri Fomin, Sergey Genkin, Ilia V. Itenberg, 1996
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