### Number 222 of a series

The radius of the small circle is 1. What is the radius of the large circles (all the same)?

Post your answer in the comments section below.

The radius of the small circle is 1. What is the radius of the large circles (all the same)?

Post your answer in the comments section below.

These are all squares with areas as marked. What is the area of the square with the question mark? Post your solution in the comments section below.

This is from a book titled *2nd Miscellany of Puzzles*. The description in the book is circuitous, so I distilled it to the essentials, as I interpret the problem. A circle sits on a 2-D grid with the coordinates marked where the circle intersects the axes. What is the diameter of the circle?

The above is a cylindrical tank with a conical top. What is the area of the top?

I haven’t worked this one. It popped up on my Facebook feed while I was lolling about on vacation, and I captured this screen shot. Here are the instructions:

- There are three patterns.
- Two diagonal and one vertical
- What goes in the center box?
- It is not mathematical, per se.
- No partial answers. Explain all three patterns.

I found this on the Internet. No fair going to Google for the answer.

Show that for any natural n, at least one of two numbers, n or n+1, can be represented in the following form: k + S(k) for a certain k, where S(k) is the sum of all digits in k. For instance, 21 = 15 + (5+1)

Post your answer in the comments section below.

Here is a calendar. It could be any calendar. I drew a 3×3 box around 9 dates. What is the sum of the numbers in the box? You are allowed five seconds.

Post your answer in the comments section below.

Mike has the right approach. To solve this in under five seconds you need to know about the life of Carl Friedrich Gauss. As a young school boy he was assigned the problem of adding up a column of large numbers. He noticed the professor wrote the column of numbers as an arithmetic series. He quickly realized the sum was the product of the average and the number of numbers. The average would be the middle number (odd number of numbers) or half way between the two middle numbers. He turned in his answer so quickly the professor was astounded.

Days in a month are an arithmetic sequence. All this problem does is to chop the sequence into three , equally spaced, sequences of the same length. The sum will be 9 times the middle number.

180 (in under 5 seconds)

I found this on the Internet, so no fair using Google.

Each of the lists below groups letters according to a certain rule. Your challenge is to find that rule and use it to determine where the X, Y, and Z would go.

List #1: A E F H I K L M N T V W

List #2: B C D G J O P Q R S U

Post your answer in the comments section below.

The circles represent four pennies arranged in a square. All right, they don’t look like pennies, but use your imagination. Also use your imagination to move two pennies to form another square smaller than the original.

Post your solution in the comments section below.

Everybody who had a go at this one got it. Here’s the picture.

What result do you get when you multiply all the digits on a phone keypad? Post your answer as a comment below.

A trick question, to be sure, and several people provided the correct answer. The trick is based on people not remembering that one of the digits on a phone key pad is a zero.

Examine the numbers in the triangular formation and determine the values for **E** and **D**. Enter your answer in the comment section below.

Mike provided an answer in the comment below. Here’s what you need to do.

First notice that 198 + 263 – 15 = 446. Then take it from there.

What’s the last digit of the following?

Enter your answer in the comments section below.

Mike saw through this one early, only his comment was so cryptic I missed it. See below. I didn’t see it until I started working through it, then I thought, “too easy.” Here is how it works.

**A** is the last digit of 17^{1999}. **B** is the last digit of 11^{1999}. **C** is the last digit of 7^{1999}.

It is obvious **A** = **C**, so the answer is **B**. And **B** is 1.

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.

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.