MA 490-505 Spring 2014
Invariants I
Parity
Problem 1 Can a 5 × 5 square checkerboard be covered by 1 × 2 dominoes?
Problem 2 Can a knight start at square a1 of a chessboard, and go to square h8, visiting each
of the remaining squares exactly once on the way?
Problem 3 A closed path is made up of 11 line segments. Can one line, not containing a
vertex of the path, intersect each of its segments?
Problem 4 Three hockey pucks, A, B, and C, lie on a playing field. A hockey player hits one
of them in such a way that it passes between the other two. He does this 25 times. Can he
return the three pucks to their starting points?
Problem 5 All the dominoes in a set are laid out in a chain (so that the number of spots on
the ends of adjacent dominoes match). If one end of the chain is a 5, what is at the other end?
Comment. A set of dominos consists of 2 × 1 rectangles with 0 to 6 spots on each square. All
28 possible pairs of numbers of spots (including doubles) are represented.
Problem 6 The number 1 through 10 are written in a row. Can the signs+ and − be placed
between them, so that the value of the resulting expression is 0?
Problem 7 The numbers 1, 2, 3, . . . , 1984, 1985 are written on a blackboard. We decide to
erase from the blackboard any two numbers, and replace them with their positive difference.
After this is done several times, a single number remains on the blackboard. Can this number
equal 0?
Problem 8 The women from Venus have five hands and the men have three. Their delegation
to the Varmland numbered 17 individuals. They decided to invite Varmlanders to a dance.
In their dance routine, everyone holds hands and together they move to the rhythm of Venusian music. For the dance to be done properly, every hand must be held by someone else’s
hand. Assuming all Varmlanders have two hands, will it be possible to do the dance with 13
Varmlanders?
Problem 9 On a paper I have written a string of 0’s and 1’s, and you can operate on this
string in three different ways:
1) The substring 01 can be replaced by 100, and vice-versa;
2) The substring 10 can be replaced by 111, and vice-versa;
3) The substring 11 can be replaced by 000, and vice-versa.
Using these rules, can you transform
a) 11001 into 0001000?
b) 0110 into 0110010?
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Problem 10 Start with the positive integers 1, . . . , 63. In one move you may replace any two
integers by their difference. Prove that an even integer will be left after 62 steps.
Problem 11 Seven pennies lie heads-up on a table. In a single move you are allowed to turn
over any four coins at the time. Using a sequence of such moves, can you get all seven coins
to lie tails-up on the table? Can it be done if you are allowed to turn over any five coins in a
single move?
Problem 12 I have filled the squares in the following grid with numbers. You may choose
any two adjacent squares (squares sharing a common side) and increase or decrease the two
numbers by the same amount. Is it possible, with a sequence of such operations, to obtain a
grid with
a) 0 in each square?
b) 1 in one square and 0 in all other squares?
c) 1 in each square of one of the diagonals and 0 in all other squares?
d) 1 in each square?
Problem 13 (a) Disks are placed in the spaces of a wheel as shown in (a). You are allowed
to add two disks to the wheel, placing one on each of two adjacent spaces. Is it possible
after a number of such movers, to get the same number of disks on each space?
(b) The same question for (b).
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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