# Pinocchio’s Hats This problem in logic from Presh Talwalkar recalled an article I wrote a while ago but did not publish.  So I thought I would post it as part of the solution.

“Assume that both of the following sentences are true:

1. Pinocchio always lies;
2. Pinocchio says, “All my hats are green.”

We can conclude from these two sentences that:

• (A) Pinocchio has at least one hat.
• (B) Pinocchio has only one green hat.
• (C) Pinocchio has no hats.
• (D) Pinocchio has at least one green hat.
• (E) Pinocchio has no green hats.”

Actually, the question is which, none or more, of statements (A) – (E) follow from the two sentences?

# A Nice Factorial Sum This is another infinite series from Presh Talwalkar, but with a twist.

“This problem is adapted from one given in an annual national math competition exam in France. Evaluate the infinite series:

1/2! + 2/3! + 3/4! + …”

The twist is that Talwalkar provides three solutions, illustrating three different techniques that I in fact have used before in series and sequence problems.  But this time I actually found a simpler solution that avoids all these.  You also need to remember what a factorial is: n! =n(n – 1)(n – 2)…3·2·1.

See a Nice Factorial Sum

# Two Candles This is another candle burning problem, presented by Presh Talwalkar.

“Two candles of equal heights but different thicknesses are lit. The first burns off in 8 hours and the second in 10 hours. How long after lighting, in hours, will the first candle be half the height of the second candle? The candles are lit simultaneously and each burns at a constant linear rate.”

See Two Candles

# Two Curious Semicircles This is a nice brain tickling problem from Presh Talwalkar.

“A circle contains two tangent semicircles whose diameters are parallel chords. If the circle has an area equal to 1, what is the combined area of the two semicircles?”

See the Two Curious Semicircles

# Three Runners Puzzle Here is another problem from Presh Talwalkar which he says is adapted from India’s Civil Services Exam.

“There are three runners X, Y, and Z. Each runs with a different uniform speed in a 1000 meters race.  If X gives Y a start of 50 meters, they will finish the race at the same time.  If X gives Z a start of 69 meters, they will finish the race at the same time.  Suppose Y and Z are in a [1000 meter] race. How much of a start should Y give to Z so they would finish the race at the same time?”

Even though Talwalkar’s original graphic showed all the runners in a 1000 meter race, it was not immediately clear to me from the wording that the race between Y and Z was also 1000 meters.  But that was the case, so I made it explicit.

See the Three Runners Puzzle

# Shy Angle Problem Here is yet another problem from Presh Talwalkar. This one is rather elegant in its simplicity of statement and answer.

“Solve For The Angle – Viral Puzzle

I thank Barry and also Akshay Dhivare from India for suggesting this problem!  This puzzle is popular on social media. What is the measure of the angle denoted by a “?” in the following diagram? You have to solve it using elementary geometry (no trigonometry or other methods).  It’s harder than it looks.  I admit I did not solve it. Can you figure it out?”

See the Shy Angle Problem.

# Sum Of Squares Puzzle Yet another interesting problem from Presh Talwalkar.

“Two side-by-side squares are inscribed in a semicircle.  If the semicircle has a radius of 10, can you solve for the total area of the two squares? If no, demonstrate why not. If yes, calculate the answer.”

This puzzle shares the characteristics of all good problems where the information provided seems insufficient.

See the Sum of Squares Puzzle.

# Hard Geometric Problem This is another problem from the indefatigable Presh Talwalkar.

_    _____Hard Geometry Problem
“In triangle ABC above, angle A is bisected into two 60° angles. If AD = 100, and AB = 2(AC), what is the length of BC?”

(Update 7/18/2020, 7/20/2020)  Alternative Solution Continue reading

# Pinwheel Area Problem Here is another engaging problem from Presh Talwalkar.

___________Triangle Area 1984 AIME
Point P is in the interior of triangle ABC, and the lines through P are parallel to the sides of ABC. The three triangles shown in the diagram have areas of 4, 9, and 49. What is the area of triangle ABC?”

See the Pinwheel Area Problem