# Problem of the Week

Four Women and a Bridge

Four women must cross a bridge over a deep ravine in enemy territoy in the middle of the night.  The treacherous bridge will hold only two women at once, and it is necessary to carry a lantern while crossing (persons must stay together while crossing).  One of the women requires 5 minutes for the trip across, one takes ten minutes, a third requires 20 minutes, and the last takes 25 minutes (each can walk slower if necessary but no faster).  Unfortunately, they have only one lantern among them.  How can they make the crossing if they have only 60 minutes before the bridge is destroyed?

The Security Guards

Keri and Mark work as security guards.  Keri has every 6th night off, and Mark has every 10th night off.  If they were both off on July 12th, what is the next day that they will both be off?

One Hundred Fives

What is the remainder when the product of one hundred 5s is divided by 7?

The Kitchen Floor

A rectangular kitchen floor is covered by square tiles.  A straight line is drawn from one corner to the corner that is diagonally opposite.

How many tiles does the line cross if the floor measures exactly 2 tiles by 3 tiles? 4 tiles by             6 tiles?  5 tiles by 7 tiles?  8 tiles by 12 tiles?  (Note: a tile is not considered to be crossed if             the line intersects it only at a corner)

Determine a rule for finding the number of tiles that the line crosses for any pair of                          whole-number dimensions of the kitchen floor.

Use your rule to determine how many tiles the line crosses if the floor measures 54 tile by              96 tiles.

An Algebraic Proof

Show that b2 + b + 1 = a2 has no positive integer solutions.

(You may work in groups of two or three and you must present the solution in class in order to receive the extra credit points.)