This is an old puzzle from Catriona Agg that I found on BL’s Math Games website.
“All four triangles are equilateral. What fraction of the largest triangle is shaded?”
This problem turned out to be both easy and unexpectedly challenging, at least for me.
See Four Equilateral Triangles for solutions.
I see another icon of the past has passed away: Tom Lehrer. Lehrer was a mathematician who put his talents to great use, in contrast with mathematicians like the neocon Paul Wolfowitz and Iraqi Ahmed Chalabi who helped foment the Iraq War. He skewered all the shibboleths of the times (1950s and 60s) with his pre-politically-correct take-downs, mainly to the music of Gilbert and Sullivan.
Among his targets was the “New Math”—the Common Core of 60 years ago and a reminder that some things never change. Fortunately, the New Math infiltrated public schools after my time, but you will recognize that its purpose is for students to understand math and not just perform rote manipulations. A truly noble intent, but its continued reincarnation indicates how difficult it is to achieve. So we are left with humor at our folly.
This video of “New Math” is from a live 1965 performance of “That Was the Year That Was”. A more famous song is Lehrer’s musical rendition of the chemical periodic table, “The Elements” (performed live in 1967 in Denmark). I invite you to listen to all his songs for an irreverent window into the past.
See “Tom Lehrer Dies at 97” for a PDF copy of his obituary.
This is another problem from the c.100AD Chinese mathematical work, Jiǔ zhāng suàn shù (The Nine Chapters on the Mathematical Art) found at the MAA Convergence website Convergence.
“A square walled city measures 10 li on each side. At the center of each side is a gate. Two persons start walking from the center of the city. One walks out the south gate, the other the east gate. The person walking south proceeds an unknown number of pu then turns northeast and continues past the corner of the city until they meet the eastward traveler. The ratio of the speeds for the southward and eastward travelers is 5:3. How many pu did each walk before they met? [1 li = 300 pu]”
See Two Men Meet for a solution.
This is an interesting problem from the 1966 Eureka magazine.
“A railway and a road run together for seven miles from P to Q. Two miles from P there is a level crossing, which is closed one minute before, and opened one minute after, a train passes.
A train passes a Stationary car at P and travels on to Q at 60 m.p.h., and, forgetting to slow down, crashes at Q; the car passes the train as it crashes. Assuming that stopping for an instant from full speed loses the car one minute, of what speed must it be capable?”
See the Railway Crossing Problem for a solution.
This is a classic puzzle from Presh Talwalkar.
“This puzzle has been asked as an interview question at tech companies like Google.
There are 100 lights numbered 1 to 100, all starting in the off position. There are also 100 people numbered 1 to 100. First, person 1 toggles every light switch (toggle means to change from off to on, or change from on to off). Then person 2 toggles every 2nd light switch, and so on, where person i toggles every ith light switch. The last person is person 100 who toggles every 100th switch.
After all 100 people have passed, which light bulbs will be turned on?”
I vaguely remembered the answer, which I confirmed after a few examples. But I didn’t remember an exact proof, so I thought I would give it a try.
See 100 Light Bulbs Puzzle for solutions.
This is a slightly challenging problem from BL’s Math Games.
“In a square garden ABCD of side 10m, a sheep sets off from B and moves along BC at 30cm per minute. At the same time, you set off from C and move along edge CD at 40cm per minute. The question is, what’s the shortest distance between you and the sheep in meters?
This is somewhat an optimization problem because as you and the sheep move along the sides of the square at different rates, the distance in between varies as you can imagine.”
There’s at least one non-calculus solution and of course one calculus solution.
See Sheep in Garden Problem for solutions.
This is a puzzle from the A+Click site.
“There is a fault with the cruise control on Hank’s car such that the speed continuously and linearly increases with time. When he starts off the speed is set to exactly 60 mph. He is driving on a long straight route with the radio on at full blast and he is not paying any attention to his speed. After 3 hours he notices that his speed has now reached 80 mph. For how many miles did he drive above the state speed limit of 70 mph?
Answer Choices: 125 miles 112.5 miles 105 miles 99.5 miles”
See Unlawful Distance for solutions.
This is another nice problem from the 2025 Math Calendar.
“The two squares in the diagram have areas 9 and 16 and form two of the sides of a triangle with area 3. What is the area x of the blue triangle?”
Again, the result must be a number of a day in a month.
See the Two Squares Puzzle for a solution.
This is another Catriona Agg puzzle that I also found somewhat challenging.
“Four squares. What’s the angle?”
Visio showed me the answer fairly soon, but it took a bit to figure out a proof.
See the Rotating Square Problem for a solution.
This is another take on the passing train type puzzle from the Moscow Puzzles.
“A train moving 45 miles per hour meets and is passed by a train moving 36 miles per hour. A passenger in the first train sees the second train take 6 seconds to pass him. How long is the second train?”
See Another Passing Train Puzzle for solutions.
