Quiz Question

Here we see a prism, for the sake of this problem let’s call it a cereal box lying flat on a table. The height of the box (lying flat) is 12 cm. The other two dimensions are 25 and 36 cm. An ant at A wants to take the shortest path to B. The ant is not allowed to crawl along the bottom of the box. How long  is the shortest path?

Post your answer in the comments section  below.

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Quiz Question

Fermat’s famous “last theorem” is illustrated above. Show that it does not hold for the following:

x = 2233445566
y = 7788990011
z = 9988776655

N is a positive integer.

Post your answer in the comments section below.

Quiz Question

Number 150 of a continuing series

This is one I got off the Internet. I left the copyright information in the image, but readers are cautioned against using that to hunt down the solution. Work this one out for yourself and post your answer in the comments section.

There are nine combinations of colored cubes pictured above. When rotated properly, two of the nine are the same. Which two are the same?

Quiz Question

Number 149 of a continuing series

Here’s one from a site on the Internet, and I’m not going to tell you what site. You need to solve this one without help.

Use the numerals 1, 9, 9 and 6 exactly in that order to make the following numbers: 28, 32, 35, 38, 72, 73, 76, 77, 100 and 1000.
You can use the mathematical symbols +, -, ×, /, √, ^ (exponent symbol) and brackets.
Example: 1×9+9×6 = 63

Post your answer in the comments section.

Quiz Question

Number 144 of a continuing series

Here is another Mensa puzzle. I ripped it right out of my copy of American Way magazine on my way to some place I had never been. The caption in the magazine says, “Supply the missing number.” I’m going to be more explicit.

The implication is that each letter A – D stands for a different number (integer). Figure out which integer each letter stands for and supply the number that goes in place of the question mark. The solution is in  the magazine, and you can still track it down. Don’t do that. Supply your answer in the comment section below.

Quiz Question

Number 143 of a continuing series

I’ve been riding on airplanes again, and I lifted this problem from the American Way magazine, courtesy of Mensa.

This isn’t a geometry puzzle. It’s a problem in mathematical logic. There is a simple logic used for the first three triangles to determine the number on the inside by applying the logic to the numbers at the vertices. Use that same logic for the fourth triangle to determine the missing number inside the triangle.

Update and solution

Yes, this really was a hard one. What you had to do was to figure out the logic that was consistent with the first three triangles. And not just any logic, but the simplest logic. And that simple logic is:

  1. Ignore the number at the top and left vertices of the triangle.
  2. Multiply the number at the right vertex by 6 to get the number in the middle.

The answer is, of course, 48.

Quiz Question

Number 142 of a continuing series

Here is another one courtesy of the Internet. See the diagram. The rectangles are identical (congruent). The perimeter of each rectangle is 222. What is the perimeter of the assembly shown above? Post your answer as a comment below.

Update and solution

This one turned out to be so easy, I’m posting the solution today. Also, I have some spare time right now waiting for Barbara Jean, and I need something to do. Here’s a helpful diagram.

It is obvious you can transform the puzzle into the form shown at the top of the above three—without altering the perimeter. Similarly for the second of the above three. Now add the piece as I have done above, and the perimeter of the resulting figure is still the same.

Each rectangle in the puzzle is h×w, height and width. The perimeter is 6h + 6w or 3 times the perimeter of a single rectangle. The answer is 666.

Quiz Question

Number 141 of a continuing series

Here is one I found on the Internet. Not much explanation was given, but I am going to assume: the numbers are the areas of the small triangles. What is the area of the remaining triangle? Post your answer as a comment below.

Update and solution

This was an easy one. To make it convenient to visualize, I have redrawn the figure above, not exactly to scale, but you should get the idea.

First I rotated the figure so the obvious line is horizontal. Now we see the problem as it is. Triangles 1 and 2 have the same altitude, h. Triangles 3 and ? have the same altitude H. Since triangle has an area of 2, its base must be twice the base of triangle 1. That means the base of triangle ? is twice the base of triangle 3. Since triangles 3 and ? have the same altitude, the area of triangle ? must be twice the area of triangle 3.

Quiz Question

One of a continuing series

This is a famous Martin Gardner puzzle. If you’re a Gardner fan, then you already know this one. Anyhow, it’s easy.

Above is a chess board. Two black squares have been removed. The task is you have dominoes, each piece being the size of two squares. Is it possible to place 31 dominoes on  the remaining squares in the chess board?

Post your answer as a comment below.

Quiz Question

One of a continuing series

Here’s one I cribbed from the Internet. It’s out there, so don’t search it out for the answer.

A group of four people has to cross a bridge. It is dark, and they have to light the path with a flashlight. No more than two people can cross the bridge simultaneously, and the group has only one flashlight. It takes different time for the people in the group to cross the bridge:

  • Annie crosses the bridge in 1 minute.
  • Bob crosses the bridge in 2 minutes.
  • Volodia Mitlin crosses the bridge in 5 minutes.
  • Dorothy crosses the bridge in 10 minutes.

How can the group cross the bridge in 17 minutes? Post your answer as a comment 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:

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467678374494482553797747268471040475346462080466842590694912933136770289891521047521620569660240580381501935112533824300355876402474964732639141992726042…

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.