### Number 209 of a series

What’s the last digit of the following?

## 17^{1999} + 11^{1999} – 7^{1999}

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What’s the last digit of the following?

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What goes in place of the ? Submit your answer as a comment below.

It would appear I started this series four years ago.

Examine the problem. The first thing that becomes apparent… All right, maybe the second thing… is the following:

3 × 15 – 1 = 14

3 × 300 – 1 = 899

So I am guessing 3 × 150 – 1 = 449, the answer.

See the diagram above. What goes in place of the question mark? Submit your answer in the comments section below.

Please note that in all cases **A** = **B** – **C** + **D**.

That’s supposed to be a chess board above, although I have not shaded in the black and white squares. It doesn’t matter, because this is not a real chess game. The black circles are black pawns, about twice as many as in a real chess game. The question is can you set a white knight on the board and proceed to capture all 16 pawns in 16 consecutive moves? Let’s assume you can, then what are the moves?

Post your answer in the comments section below.

Mike comments this is too easy. Could be. Kordemsky has this to say:

Start by capturing one of the pawns not marked with a red dot in the diagram below.

Then the rest should be straight-forward.

I’m late posting this week’s Quiz Question. Monday was a busy day. But here it is. I came across an item posted on the Internet—The Subtle Art of the Mathematical Conjecture. It has some interesting stuff, including proved and yet to be proved conjectures in mathematics. There is a discussion of Fermat’s Conjecture, and there is this:

Counterexamples can lie far ashore, like the one found by Noam Elkies, a mathematician at Harvard University, disproving Euler’s conjecture, a variation on Fermat’s conjecture that states that a fourth power can never be written as a sum of three other fourth powers. Who would have guessed that the first counterexample involved a number of 30 digits?*

The asterisk points to a note:

20,615,673

^{4}= 2,682,440^{4}+ 15,365,639^{4}+ 18,796,760^{4}.

Mike and Elmo got this one right. I had it wrong. I tried to check the math in my head and miscounted the multiplications. Shit happens.

Math time again. Some time ago I had the idea to clear out shelf space, so I dumped all the copies of Schaum’s Outline books that are available on Kindle. This is from Schaum’s *Vector Analysis*, Chapter 2.

2.11. Show that the vectors

A= −i+j,B= −i−j– 2k,C= 2j+ 2kform a right triangle.

**i**, **j**, and **k** are the unit vectors parallel to the 3D coordinate axes. To be clear, imagine each vector as a straight line oriented in space but movable. Can you move the three lines into position to form a right triangle? Give a mathematical proof.

The answer is in the book, but you know better than to look for the answer. Post your answer as a comment below.

First it is needed to demonstrate the three vectors satisfy the Pythagorean Theorem. Square the lengths of all vectors and see if the sums of two add up to the third. So we have:

|A|² = 2

|B|² = 6

|C|² =8

That works.

To form a closed polygon (triangle) the sum of the vectors must be zero.

Sum = (-1 -1)**i** (1 -1 +2)**j** (-2 +2)**k**, which is not zero. But if you reverse **A**, then

Sum = (1 -1)**i** (-1 -1 +2)**j** (-2 +2)**k**, which is zero.

Also, note the dot product of **A** and **B** is zero. The two vectors are orthogonal.

So the three vectors form a right triangle.

Here is something for those who paid attention in high school physics class. See the diagram below.

This is a metal (wood, plastic, glass) tube. The end on the left is closed, while the end on the right is open. If you blow across the open end sound waves travel back and forth within the tube. The lowest frequency at which the air column in the tube vibrates is call the fundamental frequency of the tube. In this case the fundamental frequency is 420 Hz.

The tube below is the same length as the tube above, but the left end is open. Again when you blow across the open end on the right you get the air column vibrating.

What is the fundamental frequency of the open tube? Why? Hint: wind musical instruments operate off this principle.

Post your responses in the comments section below.

Something for the math people. The plane shown above is defined by:

What is the unit vector perpendicular to the plane? Give the answer in terms of

Where i, j, and k are unit vectors for the x, y, and z axes. Submit your answer in the comments section below.

Place the numbers 1 through 19 in the circles above. The total of each three numbers along a single line must total 30.

Post your answer as a comment below. Hint, you can just list the number in the center circle and the remaining numbers as a sequence around the circle.

Here is a nice problem, not too difficult, pertinent to a current hot topic.

Hypothetical scenario: Nothing is adding carbon dioxide to the atmosphere. Carbon dioxide has a 100-year half life in the atmosphere. We crank up a contraption that pumps 100 million tons of carbon dioxide into the atmosphere each year. How much carbon dioxide is in the atmosphere when a steady state is obtained?

Post your answer as a comment below. Extra points for describing the calculation.

Yes, this is the time-honored eight queens problem. There is a star on square A2 of the chess board. Place seven more stars on the board so no two stars are on the same row, column, diagonal.

Add your solution as a comment below. Just identify the squares where the next seven stars need to go.

There are 25 numbered checkers in 25 squares. You are allowed to exchange checkers in pairs. How many exchanges are required to put all checkers in numerical order, left to right, top row to bottom row?

Post your answer as a comment below.

I was unable to solve this one, so I went to the answers in the back of the book. From page 216 (not verbatim):

The minimum solution is 19 moves.

1 -7, 7 – 20, 20 – 16, 16 – 11, 11 – 2, 2 – 24

3 – 10, 10 – 23, 23 – 14, 14 – 18, 18 – 5

4 – 19, 19 – 9, 9 – 22

6 – 12, 12 – 15, 15 – 13, 13 – 25

17 – 21

Here is another one from Facebook. What is the sum of Angles **A**, **B**, and **C**?

Post your answer in the comments section below.

The box is a square with sides = 1. The circular arcs are tangent to the sides of the square. What is the area of the colored area?

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Here is another rendition of the above. What is the area of the blue regions?

If one of the inscribed circles were complete, the square would cut out ¼ of the circle. The area of the the circle covered by the square is ¼ of the circle: π/4. The area of one blue region is 1 – π/4. The combined blue area is 2 – π/2. The area of the orange section is 1 – 2 + π/2 or **π/2 – 1**.

The image shows two circles with inscribed rectangles. The top rectangle is a square. The bottom rectangle is two squares. Are the two rectangle equal area? Describe a proof and post your answer in the comments section below.

Notice what happens when you cut the big square in half and move half of it to a new position, as I have done imperfectly here.

Above you see the well-known Tower of Hanoi puzzle. I have depicted it here as eight disks, of ascending size, stacked on an upright pole set into a game board. The object is to transfer all the disks from where they are to one of the other poles. You can move one disk at a time, and you must place it on one of the poles, on the bare board or else on a larger disk already there.

- How do you do it? What sequence of moves accomplishes the objective?
- How many moves are required?
- This shows eight disks. How long would it take, moving one disk a second, if there were 64 disks?

Post your answer in the comments section below.

This one is from *The Moscow Puzzles* by Boris A. Kordemsky, page 42.

Eight checkers are measured 1 – 8 from top down. Move one checker at a time and stack 1 – 7 from the top down on the ODD side circles, and 2 – 8 from the top down on the even side circles. Move the top checker from one pile to the top of another pile. You are not allowed to put an odd on an even checker and vice versa. You cannot put a checker on one with a lower number.

Post your answer in the comment section below.

This one is from *The Moscow Puzzles* by Boris A. Kordemsky, page 75.

Two identical boxes. Put 27 identical balls in one box. In the other box put 64 identical balls (not the same size as the first 27). All balls have the same density. Both boxes are filled to the top. In each box each layer has the same number of balls, and the outside balls in each layer touch the walls of the box. Which box contains the most weight? Also, generalize.

Post your answer in the comments section below.

209,546 is the product of two prime numbers. What are they?

344,271 is the product of two prime numbers. What are they?

Post your answer in the comments section below.

Amazing nobody supplied a correct solution, since these are so easy. Of course I picket the wording of the problem to scare people off, but look at this:

209,546 is even. It’s divisible by 2!

The digital root of 344,271 is 3, so the number is divisible by 3.

This one is from *The Moscow Puzzles* by Boris A. Kordemsky, page 75.

*Can a cube be cut with a plane to form a regular pentagon?**How about an equilateral triangle? A regular hexagon?**How about a regular polygon with more than six sides?*

Explain or show. Post your answer in the comments section below.

The answers are:

- Maybe
- Yes, yes
- Maybe

For the equilateral triangle, see the following illustration

The plane, viewed on edge, leaves a cut that is an equilateral triangle. For the regular hexagon, see this:

Place points on selected edges of the cube. The cube has 12 edges, so 1/2 the edges are selected. Place the cube in a 3-dimensional coordinate system as shown. The vector values of the points are:

A = (0, 1, 2)

B = (1, 0, 2)

C = (2, 0, 1)

D = (2, 1, 0)

E = (1, 2, 0)

F = (0, 2, 1)

A bit of vector analysis demonstrates the points are co-planar. The lines and angles are equal by symmetry argument, so the hexagon is regular.

I am thinking a regular polygon of more than six sides is not possible, but I need somebody to demonstrate this.