October Problems of the Month

Posted on October 2nd, 2018 by

Congratulations to September’s successful solvers: Filip Belik, Sam Thompson, Katherine Williams, and James Wittrig.

Throughout the fall semester, we’re hosting “Problems of the Month” here, and we invite everyone throughout the college to take part! Students, faculty, and staff are welcome to submit solutions, and there are prizes for the top student solvers during the fall (details below).

October’s Problems #1 and #2 are about colorful hexagons:

A pyramid of hexagonal cells is to be colored green, white, and/or gray. There’s a rule, though: Each cell taken together with the two cells directly below it must be either (1) all three the same color, or (2) all three different colors.

The coloring on the left is correct. The one on the right is nearly correct, but has a problem at the marked cells:

1. How many ways are there to color the six-row pyramid correctly?

2. A pyramid with 30 rows is colored according to the same rule. There is at least one green cell. What is the maximum number of cells that could be gray? (State your answer clearly, and prove that it really is the maximum.)

See the complete set of three problems here: October 2018 Problems of the Month [PDF]

The fine print: Each month’s solvers will be announced along with a running scoreboard for the Fall Series. Prizes of $125 (first place) and $50 (runner up) will be awarded to the top student solvers at the end of the Fall Series; students who have solved at least three problems during the Fall Series are eligible for the prizes. To submit your solutions:

Email solutions to jsiehler@gustavus.edu, or
Submit written solutions to Professor Siehler’s mailbox (by the door of Olin Hall 310).

 

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