Square Wheels

I came across the following problem from an Italian high school exam on the British Aperiodical website presented by Adam Atkinson:

“There have been various stories in the Italian press and discussion on a Physics teaching mailing list I’m accidentally on about a question in the maths exam for science high schools in Italy last week. The question asks students to confirm that a given formula is the shape of the surface needed for a comfortable ride on a bike with square wheels.

What do people think? Would this be a surprising question at A-level in the UK or in the final year of high school in the US or elsewhere?”

I had seen videos of riding a square-wheeled bicycle over a corrugated surface before, but I had never inquired about the nature of the surface. So I thought it would be a good time to see if I could prove the surface (cross-section) shown would do the job. See Square Wheels.

Chalkdust Grid Problem

Normally I don’t care for combinatorial problems, but this problem from Chalkdust Magazine by Matthew Scroggs seemed to bug me enough to try to solve it. It took me a while to see the proper pattern, and then it was rather satisfying.

“You start at A and are allowed to move either to the right or upwards. How many different routes are there to get from A to B?”

See the Chalkdust Grid Problem

The Weight Problem of Bachet de Méziriac

The following is a famous problem of Bachet as recounted by Heinrich Dörrie in his book 100 Great Problems of Elementary Mathematics:

“A merchant had a forty-pound measuring weight that broke into four pieces as the result of a fall. When the pieces were subsequently weighed, it was found that the weight of each piece was a whole number of pounds and that the four pieces could be used [in a balance scale] to weigh every integral weight between 1 and 40 pounds [when we are allowed to put a weight in either of the two pans]. What were the weights of the pieces?

(This problem stems from the French mathematician Claude Gaspard Bachet de Méziriac (1581-1638), who solved it in his famous book Problèmes plaisants et délectables qui se font par les nombres, published in 1624.)”

The problem has a nice solution using ternary numbers. See the Weight Problem of Bachet.

(Update 4/10/2019) Continue reading

Right Triangle with Roots

This is an interesting problem from the United Kingdom Mathematics Trust (UKMT) Senior Math Challenge of 2008.

“The length of the hypotenuse of a particular right-angled triangle is given by √(1 + 3 + 5 + … + 23 + 25). The lengths of the other two sides are given by √(1 + 3 + 5 + … + (x – 2) + x) and √ (1 + 3 + 5 + … + (y – 2) + y) where x and y are positive integers. What is the value of x + y?”

See the Right Triangle with Roots.

Turnpike Driving

This turns out to be a fairly challenging driving problem from Longley-Cook.

“Mileage on the Thru-State Turnpike is measured from the Eastern terminal. Driver A enters the turnpike at the Centerville entrance, which is at the 65-mile marker, and drives east. After he has traveled 5 miles and is at the 60-mile marker, he overtakes a man operating a white-line painting machine who is traveling east at 5 miles per hour. At the 35-mile marker he passes his friend B, whose distinctive car he happens to spot, driving west. The time he notes is 12:20 p.m. At the 25-mile marker he passes a grass cutter traveling west at 10 miles per hour. A later learns that B overtook the grass cutter at the 21-mile marker and passed the white-line painter at the 56-mile marker. Assuming A, B, the painter and the grass cutter all travel at constant speeds, at what time did A enter the turnpike?”

See Turnpike Driving.

Hyperboloid as Ruled Surface

When our daughter-in-law made wheat shocks as center-pieces for hers and our son’s fall-themed wedding reception, I naturally could not help pointing out the age-old observation that they represented a hyperboloid of one sheet. This was naturally greeted with the usual groans, but the thought stayed with me as I realized I had never proved this mathematically to myself. And so I did.

See the Hyperboloid as Ruled Surface.

(Update 10/9/2020) Spinning Rod Demo
Continue reading

Train Length Puzzle

Here is another Presh Talwalkar problem that seems unsolvable at first glance.

“Every day, a train passes a train station along a straight line track, and the train moves at a constant speed. Two friends, A and B, want to determine how long the train is. Lacking proper equipment, they devise the following method. They first synchronize their walking. Both A and B walk at the same constant speed, and each step they take is the same length. One day they line up back to back at the train station. When the front of the train reaches them, they both start walking in opposite directions. Each person stops exactly as the back of the train passes by. If person A takes 30 steps, and person B takes 45 steps, how long is the train, in terms of steps?”

See the Train Length Puzzle.

The Barrel of Beer

This is a great puzzle by H. E. Dudeney involving a very useful technique.

“A man bought an odd lot of wine in barrels and one barrel containing beer. These are shown in the illustration, marked with the number of gallons that each barrel contained. He sold a quantity of the wine to one man and twice the quantity to another, but kept the beer to himself. The puzzle is to point out which barrel contains beer. Can you say which one it is? Of course, the man sold the barrels just as he bought them, without manipulating in any way the contents.”

See the Barrel of Beer.

Pool Party

Futility Closet offers another interesting puzzle:

“A billiard ball is resting on a table that measures 10 feet by 5 feet. A player hits it with no ‘English’ and it strikes four different cushions and returns to its starting point. University of Alberta mathematician Murray Klamkin asks: How far did it travel?”

After solving the problem myself, I verified that Futility Closet provides an answer, but without real justification. So I thought I would write up my solution. See Pool Party.