Quiz Question

Number 204 of a series

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 = − ij – 2k, C = 2j + 2k form 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.

Quiz Question

Number 200 of a series

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.

 

 

Quiz Question

Number 199 of a series

This one is from The Moscow Puzzles by Boris A. Kordemsky, page 45.

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.

Quiz Question

Number 198 of a series

This one is from The Moscow Puzzles by Boris A. Kordemsky, page 43.

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.

Update and solution:

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

Quiz Question

Number 196 of a series

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?

Post your answer in the comments section below.

Update and solution:

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.

Quiz Question

Number 195 of a series

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.

Update and solution:

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.

Quiz Question

Number 194 of a series

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.

  1. How do you do it? What sequence of moves accomplishes the objective?
  2. How many moves are required?
  3. 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.

Quiz Question

Number 193 of a series

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.

Quiz Question

Number 192 of a series

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.

Quiz Question

Number 191 of a series

Back to number theory this week.

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.

Update and Solution

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.

Quiz Question

Number 190 of a series

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

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

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

Update

The answers are:

  1. Maybe
  2. Yes, yes
  3. 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.

Quiz Question

Number 188 of a series

Suppose you have a bar of metal, and you know it weighs more than 15 pound and less than 20 pounds. You have three scales (see above), and each has a maximum capacity of 5 pounds. Can you weigh the bar using the three scales? Hint: you can’t cut the bar into pieces.

Post your answer in the comments section below.

Update and Solution

Greg has already provided a workable solution. See the comment below. Here is mine. See the diagram.

The rules don’t say anything about employing extra equipment, so I have added a support block on the left and two wedge-shaped bars. I line the three scales one behind the other on the right and bridge them with a wedge. I place the other wedge on the support and lay the bar to be weighed as shown. Make sure the same amount of the bar overlaps on each end, and now the three scales support half the weight of the bar. Add the readings of the three scales to get half the weight of the bar. Account for the extra weight of the wedge.

If the bar is not uniform in thickness, then reverse the bar and do a second weighing. Sum all six scale measurements, accounting for the weight of the wedge, to get the weight of the bar.

Quiz Question

Number 186 of a series

This didn’t really happen, but imagine it. The only clock in my house is one of those grandfather types, with hands and a pendulum, and you have to wind it up. So I got up in the morning, and the clock was stopped. I forgot to wind it. I needed to set the correct time.

My friend Bill lives about five blocks away, and he always has the correct time, but no telephone. I walked over to Bill’s house and talked to him for some time. Then I walked back home and set my clock to very close to the correct time. I walked at the same rate the whole trip, but I have no idea how fast I walked.

How was I able to set my clock so accurately?

Post your answer in the comments section below.

Update and Answer

People provided answers to the Facebook posting, but nobody commented one the blog site. Anyhow, here is my answer.

 

Before I left the house I wound my clock and set the time to noon. Then I walked to Bill’s house and noted the time I arrived and the time I left. When I got home I noted how much time had elapsed on my clock, and computation of the current time was straightforward.