Quiz Question

One of a continuing series

It’s another I stole off the Internet. No fair searching for the solution.

Here are two spirals. One figure is a single blue line. The other is two blue lines. Use your eyes only—no tracing with a pencil—which one is the single line?

Post your answer as a comment below.

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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.

Quiz Question

One of a continuing series

 

Taking the easy road this week. I pulled this week’s Quiz Question from an Internet site, so don’t go searching math puzzles on Google. Copied and pasted from the site:

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: 63 = 1×9+9×6

Post your answer in the comments section below. The winner will be whoever posts the greatest number of correct solutions.

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.

ABCDEF × 3 = BCDEFA

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).

Update

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

mathematics-gemoetrydivideidentical

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.

Update

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:

XX
XXX
X

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

mathematicscomputeangle

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

Update

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:

mathematicscomputeangle-02

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

math-angleproblem

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.

Update

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

math-polygonangles

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

img_3100

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?

Stop

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.