The puzzle story for Problem 1, THE STRIPE ON BARBERPOLIA, is the seventh riddle in Martin Gardner's Riddles of the Sphinx. Here is his solution, slightly edited:

If you cut a right triangle out of paper, then roll it along its base to make a cylinder, you'll see that the hypotenuse will form a helix---a line that spirals around the cylinder from one end to the other.

Imagine that the cylinder ... has been produced in this way. In your mind, unwrap the triangle... The height of the right triangle corresponds to the cylinder's length. In this case, it is 14,400. Because the helical stripe goes seven times around the cylinder, the unwrapped triangle will have a base that is seven times the cylinder's circumference, or 7 * 8,100 = 56,700 km.

The helical stripe corresponds to the triangle's hypotenuse. Its square must equal the sum of the squares of the other two sides. The square of 14,400 is 207,360,000, and the square of 56,700 is 3,214,890,000. The sum [of these] is 3,422,250,000. A punch of your calculator's square root key gives the square root of this number as 58,500. This is the length in kilometers of the helical stripe.

As for Problems 2 and 3, EXCELSIOR is the first "Knot" of Lewis Carroll's A Tangled Tale, which was originally a magazine serial. You can find it online; there's a link from Stefan's home page. Here are Carroll's own solutions:

Problem.---"Two travelers spend from 3 o'clock till 9 in walking along a level road, up a hill, and home again: their pace on the level being 4 miles an hour, up hill 3, and down hill 6. Find distance walked: also (within half an hour) time of reaching top of hill."

Answer.---"24 miles: half past 6."

Solution.---A level mile takes 1/4 of an hour, up hill 1/3, down hill 1/6. Hence to go and return over the same mile, whether on the level or on the hill-side, takes 1/2 an hour. Hence in 6 hours they went 12 miles out and 12 back. If the 12 miles had been nearly all level, they would have taken a little over 3 hours; if nearly all up hill, a little under 4. Hence 3 1/2 hours must be within 1/2 an hour of the time taken in reaching the peak; thus, as they started at 3, they got there within 1/2 an hour of 1/2 past 6.

REFERENCES

1. Pillow Problems and A Tangled Tale, by Lewis Carroll (Charles Lutwidge Dodgson). Macmillan and Co., 1895 and 1885, respectively. Reprinted together by Dover Publications, New York. (ISBN 0-486-20493-6)
2. Riddles of the Sphinx, by Martin Gardner. The Mathematical Association of America, 1987. (ISBN 0-88385-632-8)

Department of Mathematics

Trent University