TAMU Math Circle, Spring 2012
Invariants, coloring, and graphs II, April 14
Oksana Shatalov and Igor Zelenko
1. More problems to practice the idea of remainders as invariants
Try to solve the following problems:
1. Dr. Gizmo has invented a coin changing machine which can be used in any country
in the world. No matter what the system of coinage, the machine takes any coin
and, if possible, returns exactly five others with the same total value. Prove that no
matter how the coinage system works in a given country, you can never start with
a single coin and end up with 26 coins
2. Tristan has two magic swords. One of this can cut off 21 heads of an evil Dragon.
Another sword can cut off 4 heads but after that the Dragon grows 158 new heads.
Can Tristan cut off all the heads of the Dragon, if originally they were 100 of them?
(Remark: If, for instance, the Dragon had three heads, then it is impossible to cut
them off with either of the swords.
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3. There are 13 gray, 15 brown, and 17 red chameleons on Chromatic Island. When
two chameleons of different colors meet they both change their color to the third
one ( for instance, gray and brown both become red). Is it possible that after some
time all the chameleons on the island will be of the same color?
2. More problems on graphs
4. There are 30 students in class. Can it happen that 9 of them have 3 friends each
(in class), eleven have 4 friends each and ten have 5 friends each?
5. In a certain country, 100 roads lead out of each city and one can travel along those
roads from any city to any other city. one road is closed for repair. Prove that one
can still get from any city to any other city.
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6. (a) A piece of wire is 120 cm long. Can one use it (without cutting it) to form the
edges of a cube, each of those edges is 10 cm?
(b) What is the smallest number of cuts one must make in the wire in order to be
able to form the required cube?
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