This is an initially mind-boggling problem from the 1995 American Invitational Mathematics Exam (AIME).

“Find the last three digits of the product of the positive roots of

”

See Log Lunacy for solution.

This is an initially mind-boggling problem from the 1995 American Invitational Mathematics Exam (AIME).

“Find the last three digits of the product of the positive roots of

”

See Log Lunacy for solution.

This puzzle from the Scottish Mathematical Council (SMC) Senior Mathematics Challenge seems at first to have insufficient information to solve.

“Ant and Dec had a race up a hill and back down by the same route. It was 3 miles from the start to the top of the hill. Ant got there first but was so exhausted that he had to rest for 15 minutes. While he was resting, Dec arrived and went straight back down again. Ant eventually passed Dec on the way down just half a mile before the finish.

Both ran at a steady speed uphill and downhill and, for both of them, their downhill speed was one and a half times faster than their uphill speed. Ant had bet Dec that he would beat him by at least a minute.

Did Ant win his bet?”

See the Close Race Puzzle for solutions.

(**Update 1/2/2023**) **Alternative Solution** from Oscar Rojas Continue reading

Yet another year has passed, surprisingly, with perhaps the prospect of coming out from under the shadow of the pandemic. Again, I thought I would present the statistical pattern of interaction with the website in the absence of any explicit feedback.

Perhaps due to fatigue from the height of the pandemic students seemed to have embraced returning to inclass education and abandoned online educational activities, at least as far as my website is concerned. Visits dropped precipitously this school year. Combined with a diminishing supply of fresh material this may finally spell the fading of the site. Still, I may persist if for no other reason than my own entertainment.

Anyway, here is the summary.

This is a provocative puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings for 2013.

“In the picture the top curve is a semicircle and the bottom curve is a quarter circle. Which has greater area, the red square or the blue rectangle?”

See the Spiral Areas Puzzle for solutions.

This is another physics-based problem from Colin Hughes’s *Maths Challenge* website (mathschallenge.net) that may take a bit more thought.

“A firework rocket is fired vertically upwards with a constant acceleration of 4 m/s^{2} until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 metres.

Assuming no air resistance and taking *g* = 9.8 m/s^{2}, how long does it take to reach its maximum height?”

I can never remember the formulas relating acceleration, velocity, and distance, so I always derive them via integration.

See the Fireworks Rocket for solutions.

This is a fun logic problem from the *Mathigon* math calendar for December 2022.

1. Alice said Bob did it.

2. Bob said Alice did it.

3. Carol said Alice didn’t do it.

4. Dan said it was either Alice or Carol.

Only one person is telling the truth. Who did it?

See Who Did It? for a solution.

Here is another Brainteaser from the *Quantum* magazine.

“King Arthur ordered a pattern for his quarter-circle shield. He wanted it to be painted in three colors: yellow, the color of kindness; red, the color of courage: and blue the color of wisdom. When the artist brought in his work, the king’s armor-bearer said there was more courage than wisdom on the shield. But the artist managed to prove that the proportions of both virtues were equal. Can you tell how? (A. Savin)”

This is another relatively simple problem, though it may look a bit daunting at first.

See Wisdom of Old

Alcuin of York (735-804) had a series of similar problems involving the distribution of corn among servants. Since the three propositions were the same format with only the numbers changing, I thought I would present them in a more concise form:

**“Proposition**

A certain head of household had a number of servants, consisting of men, women, and children, among whom he wished to distribute quantities, modia, of corn. The men should receive three modia; the women, two; and the children, half a modium.

(a) If the head of household has 20 servants and wished to distribute 20 modia of corn among them, let him say, he who can, How many men, women and children must there have been.

(b) If the head of household has 30 servants and wished to distribute 30 modia of corn among them, let him say, he who can, How many men, women and children must there have been.

(c) If the head of household has 100 servants and wished to distribute 100 modia of corn among them, let him say, he who can, How many men, women and children must there have been.”

I will give Alcuin’s solutions first, followed by my more expansive solutions that rely on our familiar symbolic algebra that was not available in Alcuin’s time.

This is a classic type of puzzle from Henry Dudeney.

“This is a rough sketch of the finish of a race up a staircase in which three men took part. Ackworth, who is leading, went up three steps at a time, as arranged; Barnden, the second man, went four steps at a time, and Croft, who is last, went five at a time. Undoubtedly Ackworth wins. But the point is, how many steps are there in the stairs, counting the top landing as a step?

I have only shown the top of the stairs. There may be scores, or hundreds, of steps below the line. It was not necessary to draw them, as I only wanted to show the finish. But it is possible to tell from the evidence the fewest possible steps in that staircase. Can you do it?”

See the Staircase Race for solutions.

This is a fairly straight-forward problem from Presh Talwalkar.

“A triangle is divided by 8 parallel lines that are equally spaced, as shown below. Starting from the top small triangle, color each alternate stripe in blue and color the remaining stripes in red. If the blue stripes have a total area of 145, what is the total area of the red stripes?”

See the Triangle Stripes Problem for solutions.