### Number 161 of a continuing series

### I pulled this one from the Internet. No fair searching for the solution.

Where does the hole in second triangle come from (the partitions are the same)? Post your answer in the comments section below.

Where does the hole in second triangle come from (the partitions are the same)? Post your answer in the comments section below.

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The picture says it all. What fraction of each shape is shaded? This one is on the Internet, so no fair hunting it down. Post your answer as a comment below.

I’m going to settle this week’s Quiz Question today so I can start looking for one for next week. Here are my solutions, from left to right.

This is the only one that requires some math, despite what top diagram promises. I have drawn an arrow across the width of the blue hexagon to show that it is the same width as the length of a side of the outer hexagon. It’s left to the reader to determine that the blue hexagon is 1/3 the area of the outer hexagon.

This one is easiest of them all. Slide the blue hexagon down and to the left, where I have drawn in a red hexagon of the same size. This shows that the sides of the blue hexagon are ½ the sides of the outer hexagon, so the area of the blue hexagon is ¼ the area of the outer hexagon.

A little imagination solves the last problem. I have numbered the squares 1, 2, and 3. Now rotate square number 2 45°, and you see that square 2 is ½ the area of square 1, and square 3 is ½ the area of square 2 and therefore ¼ the area of square 1.

This is from an Internet puzzle site, so don’t go searching for it on the Web. They posted it as an interactive game. You’re supposed to drag and drop the remaining numbers into the blank spaces inside the rings. Make the total inside each ring the same.

Post your answer as a comment below. You don’t have to show a picture. Just list the added numbers from left to right.

Here’s another one I got off the Internet. Which goes to show I don’t make these up. Don’t go searching the Internet for the solution.

A number of regular pentagons and squares are arranged around the outside of a large blue regular polygon. Just the lower part of the arrangement is shown below.

How many sides does the large blue regular polygon have?

Post your answer as a comment below.

Use the fact that the exterior angles of a polynomial sum to 360°. That means the exterior angles of a regular pentagon are 360/5 = 72°. That means (see the top diagram) the exterior angles of the blue polygon are 90 – 72 = 18°. The exterior angles of blue polygon must total 360, so the blue polygon must have 20 sides.

Each color in the above diagram represents a separate piece of a puzzle. Rearrange the pieces to form the figure below.

The images are to scale, so you can print the top one and cut the pieces apart to work the puzzle. Scan your solution and post the image as your solution. Use the comments section below.

Two people have proposed solutions. I printed the puzzle out and cut out the pieces. I was unable to solve it sitting at the breakfast table, I was when I went to the puzzle’s Web site and used their interactive controls to manipulate the pieces. Here is what I came up with.

Here is Helen’s solution.

Mike proposed a solution. See the comment below.

Three people are gathered in a room, and the puzzle master sticks a number on the forehead of each. Each can see the numbers on the foreheads of the other two, but nobody can see what’s on his own forehead. Here are the conditions of the three numbers, A, B, And C.

- A + B = C
- A > 0
- B > 0
- C > 0
- All the numbers are different.

All there contestants are have perfect logic, and all know this fact and also the conditions stated above..

The puzzle master turns to person number 1 and says, “Look at the other two, and tell me what number is on your forehead.” Person 1 looks at the other two, and he sees 20 on one and 30 on the other. He says he is unable.

Same for person 2. He is unable.

Same for person 3. He is unable.

The puzzle master returns to person 1. He announces what number he has on his forehead. What is that number? Post your answer in the comments section below.

I have ten boxes which I want to pack into crates. Each crate can carry a maximum of 25 kg.

But I only have three crates, and the total weight of the boxes is 75kg:

15 kg, 13kg, 11 kg, 10 kg, 9 kg, 8 kg, 4 kg, 2 kg, 2kg, 1 kg

How can I pack the boxes into the crates?

Post your solution as a comment below.

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.

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.

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.

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.

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.

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.

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:

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

The answer is, of course, 48.

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.

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.

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.

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.

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.

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.

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.

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.

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.

AB^{2}=AC^{2}+ CB^{2}

(x + 2)^{2}= (2 – x)^{2}+ 4

x = ½

See the image. The circles are radius 2.5 and 1.5. What is the area of the red section?

Post your answer as a comment below.