# Circular Rendezvous Mystery

Here is yet another surprising result from Colin Hughes at Maths Challenge.

Problem
It can be shown that a unique circle passes through three given points. In triangle ABC three points A’, B’, and C’ lie on the edges opposite A, B, and C respectively. Given that the circle AB’C’ intersects circle BA’C’ inside the triangle at point P, prove that circle CA’B’ will be concurrent with P.”

I have to admit it took me a while to arrive at the final version of my proof. My original approach had some complicated expressions using various angles, and then I realized I had not used one of my assumptions. Once I did, all the complications faded away and the result became clear.

# Trains – Pickleminster to Quickville

This is another train puzzle by H. E. Dudeney. This one has some hairy arithmetic.

“Two trains, A and B, leave Pickleminster for Quickville at the same time as two trains, C and D, leave Quickville for Pickleminster. A passes C 120 miles from Pickleminster and D 140 miles from Pickleminster. B passes C 126 miles from Quickville and D half way between Pickleminster and Quickville. Now, what is the distance from Pickleminster to Quickville? Every train runs uniformly at an ordinary rate.”

# Mountain Houses Problem

It is always fascinating to look at problems from the past. This one, given by Thomas Whiting himself, is over 200 years old from Whiting’s 1798 Mathematical, Geometrical, and Philosophical Delights:

Question 2, by T. W. from Davison’s Repository.
There are two houses, one at the top of a lofty mountain, and the other at the bottom; they are both in the latitude of 45°, and the inhabitants of the summit of the mountain, are carried by the earth’s diurnal rotation, one mile an hour more than those at the foot.

Required the height of the mountain, supposing the earth a sphere, whose radius is 3982 miles.”

See the Mountain Houses Problem

# Consecutive Product Square

This problem from Colin Hughes at Maths Challenge is a most surprising result that takes a bit of tinkering to solve.

Problem
We can see that 3 x 4 x 5 x 6 = 360 = 19² – 1. Prove that the product of four consecutive integers is always one less than a perfect square.”

The result is so mysterious at first that you begin to understand why the ancient Pythagoreans had a mystical relationship with mathematics.

See the Consecutive Product Square.

# Impossible Car Riddle

This is another intriguing problem from Presh Talwalkar.

“A car travels 75 miles per hour (mph) downhill, 60 mph on flat roads, and 50 mph uphill. It takes 3 hours to go from town A to B, and it takes 3 hours and 30 minutes for return journey by the same route. What is the distance in miles between towns A and B?”

See the Impossible Car Riddle

# Lucky 7 Problem

This interesting problem comes from Colin Hughes at the Maths Challenge website.

Problem
Prove that for any number that is not a multiple of seven, then its cube will be one more or one less than a multiple of 7.”

# Tandem Bicycle Puzzle

A glutton for punishment I considered another Sam Loyd puzzle:

“Three men had a tandem and wished to go just forty miles. It could complete the journey with two passengers in one hour, but could not carry the three persons at one time. Well, one who was a good pedestrian, could walk at the rate of a mile in ten minutes; another could walk in fifteen minutes, and the other in twenty. What would be the best possible time in which all three could get to the end of their journey?”

See the Tandem Bicycle Puzzle.

# Cube Slice Angle Problem

This is from the UKMT Senior Challenge of 2004.

“L, M, and N are midpoints of a skeleton cube, as shown. What is the value of angle LMN?

_____A_90°_____B_105°_____C_120°_____D_135°_____E_150°”

# Marching Cadets and Dog Problem

In my search for new problems I came across this one from Martin Gardner:

“A square formation of Army cadets, 50 feet on the side, is marching forward at a constant pace [see Figure]. The company mascot, a small terrier, starts at the center of the rear rank [position A in the illustration], trots forward in a straight line to the center of the front rank [position B], then trots back again in a straight line to the center of the rear. At the instant he returns to position A, the cadets have advanced exactly 50 feet. Assuming that the dog trots at a constant speed and loses no time in turning, how many feet does he travel?”

Gardner gives a follow-up problem that is virtually impossible:

“If you solve this problem, which calls for no more than a knowledge of elementary algebra, you may wish to tackle a much more difficult version proposed by the famous puzzlist Sam Loyd. Instead of moving forward and back through the marching cadets, the mascot trots with constant speed around the outside of the square, keeping as close as possible to the square at all times. (For the problem we assume that he trots along the perimeter of the square.) As before, the formation has marched 50 feet by the time the dog returns to point A. How long is the dog’s path?”

See the Marching Cadets and Dog Problem.

# Star Polygon Problem

This is problem #25 from the UKMT 2014 Senior Challenge.

“Figure 1 shows a tile in the form of a trapezium [trapezoid], where a = 83⅓°. Several copies of the tile placed together form a symmetrical pattern, part of which is shown in Figure 2. The outer border of the complete pattern is a regular ‘star polygon’. Figure 3 shows an example of a regular ‘star polygon’.
How many tiles are there in the complete pattern?
_____A_48_____B_54_____C_60_____D_66_____E_72”

See the Star Polygon Problem.