Lorentz Transformation

Over the years one of the subjects I return to periodically to study is Einstein’s Theory of Relativity, both the Special and General theories. Interest in the Special Theory focused on the derivation of the Lorentz transformations (or contractions). Why did objects appear with different lengths and clocks run at different speeds for observers moving relative to one another? Early on (late 60s) I came across a great explanation in the 1923 book by C. P. Steinmetz. He derived it from two general assumptions of special relativity: (1) that all motion is relative, the motion of the railway train relative to the track being the same as the motion of the track relative to the train, and (2) that the laws of nature, and thus the velocity of light, are the same everywhere. I did not follow his derivation completely, so I produced my own, which I will give here. See the Lorentz Transformation.

Star Sum of Angles

This problem posted by Presh Talwalkar offers a variety of solutions, but I didn’t quite see my favorite approach for such problems. So I thought I would add it to the mix. “Thanks to Nikhil Patro from India for suggesting this! What is the sum of the corner angles in a regular 5-sided star? What is a + b + c + d + e = ? Here’s a bonus problem: if the star is not regular, what is a + b + c + d + e = ?” See Star Sum of Angles

More Pool

This is another UKMT Senior Challenge problem, this time from 2006. “A toy pool table is 6 feet long and 3 feet wide. It has pockets at each of the four corners P, Q, R, and S. When a ball hits a side of the table, it bounces off the side at the same angle as it hit that side. A ball, initially 1 foot to the left of pocket P, is hit from the side SP towards the side PQ as shown. How many feet from P does the ball hit side PQ if it lands in pocket S after two bounces?” Pool Partiers should have no difficulty solving this. See More Pool.

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.

f(x) = √2 – (e^x + e^-x)/2

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.

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.