Quiz Question

One of a continuing series

Continuing into the new year, here are two easy geometry problems.

Compute the perimeters of the two shapes. Enter your answer in the comments section below.


Quiz Question

One of a continuing series

Merry Christmas. Here is an easy one. Make the usual assumptions from the drawing. What is the value of x? Post your answer as a comment below.

Update and solution

I expected somebody would solve this quickly, and Mike nailed it within hours of posting. Mike only provided the solution. See the comment below. Here is how it unravels. See the image.

Obviously this is a semicircle and a square with a line tangent to the circle. We now have a right triangle ABC, tangent to the circle at D.

From  basic geometry we know that DB = 2. Also x = EA = AD. From there everything falls out quickly.

AB2 =AC2 + CB2

(x + 2)2 = (2 – x)2 + 4

x = ½

Quiz Question

One of a continuing series

Keeping with a run of math questions… This problem is on the Internet. You have to provide an answer without going to the Internet.

The large arc is centered at O, The small arc is centered at D. Prove the two shaded areas are equal.

Post  your answer as a comment below.

Update and solution

Mike and Steve have provided correct solutions. See the comments. Steve worked out the math, and Mike stated the path to resolution rather cryptically. Both invoked π, which is not necessary. Try this approach.

The triangle is a right, equilateral triangle. The hypotenuse is √2 times the base and is also the diameter of the small semicircle. You will have no problem from that point concluding the small semicircle’s area is ½ the area of the large semicircle. The area A of the small semicircle is equal to the area of the triangle + the circle segment subtended by the triangle’s hypotenuse. The area of the triangle is A – the area of the segment. The area of the lune outside the large semicircle is A – the area of the segment. Therefore the two areas are the same.

Quiz Question

One of a continuing series

Back to math questions for a change. Full disclosure: I don’t make up all of these. This is from an Internet site. No fair going to the Internet to get the answer.

The triangle is equilateral. Prove the shaded area is equal to the inner circle. Post your answer as a comment below.

Update and solution

Mike is the first and only to provide the correct solution. A reasoning goes like this.

It is easy to demonstrate (exercise left to the reader) that the inner circle is ¼ the area of the outer circle. Then the region between the inner and outer circles is ¾ the area of the outer circle. The blue-shaded regions total 1/3 of this difference or ¼ the area of the outer circle. The inner circle is equal to the blue-shaded area.

Bad Movie Wednesday

One of a continuing series

This one has been hanging out on Amazon Prime Video for a while, and this week (July) I decided to give it a look. Interesting thing is I didn’t watch it on the big TV, just brought it up on my computer and sat through the showing at my desk. It’s π, as spelled in the title or rather Π if you want to capitalize it. As you can guess, there’s going to be some math involved. It came out nearly 20 years ago (1998). Details are from Wikipedia.

It’s a hodge-podge of images and scenes, and it’s in monochrome. Think Last Year At Marienbad brought forward 37 (now 56) years, and substitute technology and math for sex and social conflict, and you get the idea. Only Pi doesn’t have all that endless repetition. I will show some screen shots and skim the plot.

Max Cohen (Sean Gullette) lives alone, and he’s a mental aberration. Due to an early medical convulsion his brain is an organic computer, and he performs amazing feats of mental calculation and sees (or at least looks for) patterns everywhere, including within the decimal representation of pi. In fact, that’s how the movie starts out:


And some more. You get the idea.

Max lives alone, and he’s built this rude computer that does amazing things, although I was never able to figure out by watching the movie what made the computer so special. There is talk about finding patterns everywhere, including in π.

Hold it right there. Full disclosure: I have a college degree in mathematics, and my information is there is good logic to conclude π and other irrational numbers do not contain any patterns. Irrational numbers are numbers that are not the quotient of two integers, and this includes numbers like √2, √3, √5, √10, cube root of 15, and so on. Irrational numbers also include the transcendental numbers, such as π, the natural logarithm of 6, the sine of a 61° angle, and also the ever popular e. There are no patterns. But that’s what this movie is all about.

Max spends his days getting  the computer to spit out a special number, which significance I was never able to determine during one watch-through. He sits at a lunch counter and scribbles numbers on stock market listings. He has the idea the fluctuations of the market have a deeply-embedded pattern he will be able to deduce, once he has solved his riddle.

He is constantly beleaguered by migraine headaches and convulsions, requiring periodic dosing and injections.

At the lunch counter Max is besieged by Lenny Meyer (Ben Shenkman) a Jew (Max is a non-religious Jew), who pries into what Max is doing.

He plays go with his friend and mentor Sol Robeson (Mark Margolis). Sol urges Max to quit the hopeless quest. There is no pattern. Max’s work borders on numerology.

Max interacts with neighbors in his apartment building, one being Devi (Samia Shoaib) the woman who lives next door and who flirts with him. He pays her no mind.

A big concern is to predict the stock market, a goal of many and a factor that brings intrigue and danger into Max’s life.

A persistent woman, Marcy Dawson (Pamela Hart), keeps trying to get face time with Max to a point he can no longer put her off. Things begin to take on a sinister tone.

Marcy’s friends offer Max the use of a special computer chip, and he uses it to recompute a 216-digit number he previously produced and then threw away. This number is the secret to predicting the stock market. Max is unable to print the number, but he has memorized it. Marcy and her friends want the number, and they put the squeeze on Max.

Lenny’s Jewish friends rescue Max and attempt to force him to reveal the number. It will unlock the secrets of the Torah and restore the Ark of the Covenant. Max refuses to cooperate.

He despairs of the whole business and uses an electric drill to perform a trepanning on his head, since shaved.

At the end, in the park, when the young girl living in his building asks him for various mathematical computations, he is unable to do them, while she performs the operation using her hand calculator.

There are parts I left out, mostly stuff I didn’t understand, such as the squishy thing Max finds on the steps in the subway, said squishy thing that responds amazingly when Max prods it.

This is all about number theory. There no sets, no cosets, no differential equations, and no topological congruencies. There’s lots about Fibonacci series and spirals and golden ratios. Probably a semester’s worth of pure math is lost somewhere in here.

Quiz Question

One of a continuing series

This is from somebody else. It showed up on my Facebook feed just in  time, when I needed inspiration for a new Quiz Question. It’s easy. Give yourself about 15 seconds. The problem was posed as:

There are three boxes and three statements. There is a car in only one of the boxes. Only one statement is true. Which statement is true, and in which box is the car?

Post your answer as a comment below.

Quiz Question

One of a continuing series

Got this one from the Internet, so no fair going to Google for the answer.


Substitute a digit for each letter to provide the correct equation. Post your answer as a comment below. The solution will be provided next week (or sooner).


No solution given yet. I have not taken the time to solve this, but here are some hints.

Note that A < 4 and A ≠ 0. A ≠ 0 is not stated in the problem, but I’m taking it as assumed. If A > 3, then multiplying by three would produce overflow and a number with more digits.

BCDEFA is divisible by 3, which means ABCDEF is divisible by 3, since both have the same digital root.

BCDEFA is divisible by 9.

That should get people going, so I’m going to give more time to come up with an answer.

Quiz Question

One of a continuing series

Easy one for a change, so give yourself 10 seconds to work it. It’s a single water hose with the ends uncoupled. Where are the ends?

Post your answer in the comment section below.

Update and solution

The solution is straight-forward. See the revised picture below.

Draw circles (ellipses) around A and B. Each has three hoses crossing into (or out of) the ellipse. Therefore, there must be a hose end within each of the two ellipses. Since there are only two ends (one hose), the ends must be under A and B. You don’t need to examine C and D, but if you do you will observe an even number of crossings.

Quiz Question

One of a continuing series


Readers have been getting off easy recently. I’m going back to geometry questions, so give your brain a work out.

I found this on the Internet, but you shouldn’t go looking for the solution without first coming up with a solution. With a single line, does not need to be straight, divide the shape shown above into two identical parts. Post your answer as a comment below.

Actually, send me a copy of your solution by email, and I will post it.


No solution. I have not solved it. Mike proposed a solution. See his comment below. Unable to post a graphic, he indicated the shape of the solution as follows:


See the figure below:

Shape A is the original, turned upright. Shape B is Mike’s proposed solution in graphical form. My apologies if I misinterpreted Mike’s rendition.

What is apparent to me is that shape B cannot be fitted twice into shape A. I’m calling the Quiz Question still  unanswered.

Quiz Question

One of a continuing series


Mathematics again. What is the value of the indicated angle? Post your answer as a comment below.


Three people have submitted correct answers, all holding Ph.D. degrees, but none in mathematics. Here is my solution, which I believe to be the simplest approach. See the diagram:


It’s the same as the original diagram, but I have added some labels, and I have added line BC.

Notice immediately that BC is the same as AB. If you don’t notice this immediately, then stop reading now and get into another line of work. Now notice that angle BAD is the same as angle EBC. Again, if you don’t notice, stop reading. BAD = EBC implies ABC is a right angle. Again, you can quit while you’re ahead. We have a right isosceles triangle, which means that BAC is 45°. And no mental gymnastics have been required.

Quiz Question

One of a continuing series


This popped up on my Facebook time line, posted by somebody else. So I stole it, and here it is: What is the sum of all the blue angles? Post your answer as a comment below.


A number of people have posted responses, so I am going to supply the answer. See the following diagram:


What is the sum of interior angles of a polygon? The example of a triangle explains. The triangle is ABC, defined by its three interior angles. But concentrate on the complementary angles a and b and c. What is the sum of those angles? Consider the line ab. Line bc branches off from ab with a change of direction equal to angle b. Follow the path around the triangle, and the total change of direction is 360 degrees. That’s going to be the total of a and b and c. The sum of a and A is 180° so the sum of all angles is 3 × 180 = 540. 540 – 360 = 180, the sum of interior angles of all triangles.

The method holds true for all polygons. The polygon in this puzzle is unusual in that the path makes two complete turns or 360 × 2 = 720. There are 6 sides and six interior angles, so the sum of the interior angles is 6 × 180 – 720 = 360.

Quiz Question

One of a continuing series


This is going to be very easy for most. Some will trip up on it.

There are three closed boxes. One box has only apples. One has only bananas. One has a mixture of apples and bananas. Each box is labeled to identify the contents. The problem is, all the labels are wrong. You are allowed to peek inside one box and are required to determine the contents of the two remaining boxes. Which box do you open and look into?


If you have gotten to this point you have gone too far. You should not have to think about this problem to solve it. Just post your answer in the comments section below. And also tell everybody why this didn’t require working through the possibilities to come up with the correct answer.

Quiz Question

One of a continuing series


This week’s Quiz Question should be easy. Easy for those who stayed awake in class. As I understood it, this was explained once and then never again.

Myriad is a number. What number? How much is a myriad?

You can look this up on the Internet, but don’t. Enter your answer as a comment below.


Greg wins. His is the correct answer. A myriad is 10,000. How this name came about, I do not know. It’s time to look it up:

myriad (n.) 1550s, from Middle French myriade and directly from Late Latin myrias (genitive myriadis) “ten thousand,” from Greek myrias (genitive myriados) “a number of ten thousand, countless numbers,” from myrios (plural myrioi) “innumerable, countless, infinite; boundless,” as a definite number, “ten thousand”

Greg was paying attention in grade school.

Quiz Question

One of a continuing series


I stole this from somebody else:

Continue the following number series with the group of numbers below which best continues the series?

1 10 3 9 5 8 7 7 9 6 ? ?

11 5
10 5
10 4
11 6

Provide your answer in the comments section.

Update and Answer

Take a look at the sequence. The first number and every second thereafter increases by 1. The second number, and every second thereafter decreases by two. Therefore the next two number in the sequence are 10 and 4.

Update and Correction

Actually, I got it backward. It’s the odd numbers that increase by 2, and it’s the even numbers that decrease by 1.