A Mathematician in the Jury Box

This is a great posting by the mathematician James Propp on his website Mathematical Enchantments, not only for the main story about his experience as a mathematician at a trial, but also for his short excursion into the idea of definitions in mathematics—basically the same topic I was trying to address in my posts on “A Meditation on ‘Is’ in Mathematics” here and here.

Normally I would excerpt such an article and provide the link to the full article, but given the short half-life of links I have also provided a full PDF copy just in case. You should visit the link to see the comments, which I have not included (Propp is lucky to have some of those). In fact, you should take the opportunity to peruse a number of his articles, if you have not already, which are longer in nature, appear monthly, and provide some great insights into mathematics (his pseudosphere icon is the Beltrami surface, which he discusses elsewhere).

See A Mathematician in the Jury Box.

Rising Sun

Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2012:

“A semicircle of radius r is drawn with centre V and diameter UW. The line UW is then extended to the point X, such that UW and WX are of equal length. An arc of the circle with centre X and radius 4r is then drawn so that the line XY is tangent to the semicircle at Z, as shown. What, in terms of r, is the area of triangle YVW?”

See the Rising Sun

Six Squares Problem

This is a problem from the UKMT Senior Challenge for 2001. (It has been slightly edited to reflect the colors I added to the diagram.)

“The [arbitrary] blue triangle is drawn, and a square is drawn on each of its edges. The three green triangles are then formed by drawing their lines which join vertices of the squares and a square is now drawn on each of these three lines. The total area of the original three squares is A1, and the total area of the three new squares is A2. Given that A2 = k A1, then

_____A_ k = 1_____B_ k = 3/2_____C_ k = 2_____D_ k = 3_____E_ more information is needed.”

I solved this problem using a Polya principle to simplify the situation, but UKMT’s solution was direct (and more complicated). See the Six Squares Problem.

The Damaged Engine

Yet another train problem from H. E. Dudeney.

“We were going by train from Anglechester to Clinkerton, and an hour after starting an accident happened to the engine. We had to continue the journey at three-fifths of the former speed. It made us two hours late at Clinkerton, and the driver said that if only the accident had happened fifty miles farther on the train would have arrived forty minutes sooner. Can you tell from that statement just how far it is from Anglechester to Clinkerton?”

See the Damaged Engine.

Triangle Acute-Angle Problem

Catriona Shearer retweeted the following problem from Antonio Rinaldi‏ @rinaldi6109

“My little contribution to @Cshearer41 October 7, 2018
A point D is randomly chosen inside the equilateral triangle ABC. Determine the probability that the triangle ABD is acute-angled.”

See Triangle Acute-Angle Problem

Parallelogram Cosine Problem

Another challenging problem from Presh Talwalkar. I certainly could not have solved it on a timed test at the age of 16.

One Of The Hardest GCSE Test Questions – How To Solve The Cosine Problem

Construct a hexagon from two congruent parallelograms as shown. Given BP = BQ = 10, solve for the cosine of PBQ in terms of x.

This comes from the 2017 GCSE exam, and it confused many people. I received many requests to solve this problem, and I thank Tom, Ben, and James for suggesting it to me.”

See the Parallelogram Cosine Problem

Hitting the Target

Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2012:

“Tom and Geri have a competition. Initially, each player has one attempt at hitting a target. If one player hits the target and the other does not then the successful player wins. If both players hit the target, or if both players miss the target, then each has another attempt, with the same rules applying. If the probability of Tom hitting the target is always 4/5 and the probability of Geri hitting the target is always 2/3, what is the probability that Tom wins the competition?

______A 4/15______B 8/15______C 2/3______D 4/5______E 13/15”

See Hitting the Target