In my search for problems I decided to purchase Dan Griller’s GCSE problem book mentioned in the Cube Roots Problem. I am still a bit confused about the purpose of the GCSE exam and who it is for, since the other problems in Griller’s book are often as challenging or more so than the cube roots problem. It is hard to believe students not pursuing college level degrees could solve these problems. (Grades 8 and 9 referred to in the subtitle of the book must indicate something other than US grades 8 and 9, since the exams are aimed at 16 year-olds, not 13 and 14 year-olds.)
Supposedly the problems in Griller’s book are nominally arranged in increasing order of difficulty from problem 1 to problem 75. However it seemed to me that there were challenging problems scattered throughout and the last problem was not all that much harder than earlier ones. And many of them had a whiff of Coffin Problems—they seemed impossible at first (Problem 44: Construct a 67.5° angle!). I don’t know how many problems are on the exam or how long the exam is, but anyone taking a timed exam does not have the leisure to mull over a problem. The student only has a few minutes to come up with an approach and clever insights are rare under the circumstances. Anyway, here is the last problem in the book.
“Problem 75. A square pond of side length 2 metres is to be surrounded by twelve square paving stones of side length 1 metre.
(a) The first design is constructed with a circle whose centre coincides with the centre of the pond. Calculate exactly the total dark grey area for this design.
(b) The second design is similar. Calculate exactly the total dark grey area for this second design.”
See the Pool Paving Problem
This is a problem from the UKMT Senior Challenge for 2019. (It has been slightly edited to reflect the colors I added to the diagram.)
“The edge-length of the solid cube shown is 2. A single plane cut goes through the points Y, T, V and W which are midpoints of the edges of the cube, as shown.
What is the area of the cross-section?
See Another Cube Slice Problem
This is a nice Brainteaser from the Quantum math magazine.
“Line segment MN is the projection of a circle inscribed in a right triangle ABC onto its hypotenuse AB. Prove that angle MCN is 45°.”
See the Circle Projection Problem.
I found an interesting geometric statement in a paper of Glen Van Brummelen cited in the online MAA January 2020 issue of Convergence:
“For instance, Abū’l-Wafā’ describes how to embed an equilateral triangle in a square, as follows: extend the base GD by an equal distance to E. Draw a quarter circle with centre G and radius GB; draw a half circle with centre D and radius DE. The two arcs cross at Z. Then draw an arc with centre E and radius EZ downward, to H. If you draw AT = GH and connect B, H, and T, you will have formed the equilateral triangle.”
So the challenge is to prove this statement regarding yet another fascinating appearance of an equilateral triangle.
See the Triangle of Abū’l-Wafā’
This turned out to be a challenging geometric problem from Poo-Sung Park posted at the Twitter site #GeometryProblem
“Geometry Problem 92: What is the ratio of a:b?”
See the Envelope Puzzle
Here is another problem from the Quantum magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).
“A quadrangle is inscribed in a parallelogram whose area is twice that of the quadrangle. Prove that at least one of the quadrangle’s diagonals is parallel to one of the parallelogram’s sides. (E. Sallinen)”
See Quadrangle in Parallelogram
Here is another Brain Bogglers problem from 1987 by Michael Stueben.
“A quadrilateral with sides three, two, and four units in length is inscribed in a circle of diameter five. What’s the length of the fourth side of the quadrilateral?”
Like a number of other Brain Bogglers this problem also uses an insight that makes the solution easy.
See the Quad in Circle Problem.
Here is another Brainteaser from the Quantum magazine.
“Prove that the area of the red portion of the star is exactly half the area of the whole star. (N. Avilov)”
This is a relatively simple problem, but I wanted to include it because of its cartoon. Its implied gentle post-Soviet humor reminded me of that strange decade in US-Russian affairs between the end of the Cold War and the rise of Putin in the 21st century. The strangeness was brought home when we had our annual security checks of our classified document storage. Being mostly anti-submarine warfare (ASW) material the main concern was that it would not fall into the hands of the Soviets. But with the “demise” of the Soviet Union in 1989 no one cared any more about the classification. After decades of painfully securing these documents we could not suddenly turn them loose and throw them into the public trash. So we kept them secure anyway. You can imagine how we old cold-warriors feel about the current regime.
That is not to say that I didn’t welcome the thaw. Russian literature, both classical and even “Soviet realism”, as well as Russian cinema, is some of the world’s best. And Russian mathematicians have always been superior, and especially adept at communicating with novices. The collaboration of the American mathematicians and Kvant contributors in Quantum produced excellent results during the thaw. It is unfortunate that it could not survive the rise of Putin and his oligarchs.
See the Red Star
Here is another UKMT Senior Challenge problem from 2017, which has a straight-forward solution:
“The diagram shows a circle of radius 1 touching three sides of a 2 x 4 rectangle. A diagonal of the rectangle intersects the circle at P and Q, as shown.
What is the length of the chord PQ?
__A_√5____B_4/√5____C_√5 – 2/√5____D_5√5/6____E_2”
See the Circle in Slot Problem
Having fallen under the spell of Catriona Shearer’s geometric puzzles again, I thought I would present the latest group assembled by Ben Orlin, which he dubs “Felt Tip Geometry”, along with a bonus of two more recent ones that caught my fancy as being fine examples of Shearer’s laconic style. Orlin added his own names to the four he assembled and I added names to my two, again ordered from easier to harder.
See Geometric Puzzle Munificence.
(Update 4/16/2020) Ben Orlin has another set of Catriona Shearer puzzles 11 Geometry Puzzles That Drive Mathematicians to Madness which I will leave you to see and enjoy. But I wanted to emphasize some observations he included that I think are spot on. Continue reading