Quiz Question

Number 231 of a series

 

There are two treasure chests, both closed. One chest contains 100 gold coins. The other contains 50 gold coins and 50 silver coins.

You randomly chose a chest to open, and you do not look inside, but you remove a coin and close the lid. You look at the coin in your hand, and it is gold. What is the probability you chose the chest with 100 gold coins?

Post your answer in the comments section below.

Bad Joke of the Week

One of a Continuing Series

A selection especially for math lovers

2. Why do teenagers travel in groups of 3 or 5?

Because they can’t even.

3. Why should you worry about the math teacher holding graph paper?

She’s definitely plotting something.

6. Why is it sad that parallel lines have so much in common?

Because they’ll never meet.

7. Are monsters good at math?

Not unless you Count Dracula.

8. Why are obtuse angles so depressed?

Because they’re never right.

9. What’s the best way to woo a math teacher?

Use acute angle.

10. Did you hear about the mathematician whose afraid of negative numbers?

He’ll stop at nothing to avoid them.

11. How come old math teachers never die?

They tend to just lose some of their functions.

12. My girlfriend is the square root of -100.

She’s a perfect 10, but purely imaginary.

13. How do you stay warm in any room?

Just huddle in the corner, where it’s always 90 degrees.

14. Did you hear the one about the statistician.

Probably.

15. What’s the best way to serve pi?

A la mode. Anything else is mean.

16. A farmer counted 297 cows in the field.

But when he rounded them up, he had 300.

This page has a total of 56 of these, that is if numbers matter.

Quiz Question

Number 228 of a series

Shown here is a cube that fits snugly inside a sphere. It may not appear so from this diagram, but all eight corners of the cube touch the surface of the sphere. The length of each edge of the cube is 8. What is the diameter of the sphere?

It may not appear so, but the solution is ridiculously easy. Post your solution in the comments section below.

Update and Solution

This looks hard until you remember the Pythagorean Theorem applies to spaces of all dimensions. I came upon this when a problem at work required computing the distance between two points in a 10-dimensional space.

The diagonal of a rectangle is the square root of the sum of the squares of the two sides. The diagonal of a prism is the square root of the sum of the squares of the three dimensions.

Since the diagonal of the cube is the diameter of the circle, the diameter is 8√3.

Quiz Question

Number 226 of a series

The diameter of the small circles is 2. What is the diameter of the large circle.

Post your answer in the comments section below.

Update and Solution

See the diagram below.

The centers of the three small circles are at the apexes of equilateral triangle ABC. All that is necessary is to determine the distance from C to the center of the triangle. Since the sides of the triangle are 2, then the distance to the center is two thirds of 2(½√3) ≅ 1.155. So the radius of the large circle is 1 + 1.155 approximately, and the diameter is 4.309 approximately.

Quiz Question

Number 225 of a series

See the two diagrams above. The three large spheres have a diameter of 3. The small sphere has a diameter of 1. The small sphere is stacked in the gap between the three large spheres. What is the value of h?

Post your answer in the comments section below.

Update and Solution

See the following diagrams.

The centers of the spheres are at the vertices of a quadrahedron whose base is an equilateral triangle. We have AB = BC = CA = 3, and  AD = BD = CD = 2. What is the height of the quadrahedron?

The base of the tetrahedron is an equilateral triangle with each side being 3. The altitude of this triangle is 3(√3)/2. The height of the tetrahedron is the altitude of a right triangle with hypotenuse = 2 and base = (2/3) the altitude of the equilateral triangle. So the height of the tetrahedron is 1.

So h = 1 + radius of large sphere + radius of small sphere = 3. It’s early in the morning, so you may want to check my math.

Quiz Question

Number 224 of a series

The diagrams above are an overhead view and a front view of two spheres backed into a corner. The drawings are not to scale, so believe me when I tell you the radius of the small sphere is 1, and the radius of the large sphere is 4. What is the value of h?

Post your answer in the comments section below.

Update and Solution

See the following two diagrams.

The radius of the small sphere is 1, so d = √2. Similarly for the large sphere, radius = 3.

We get b = 3√2 – √2 = 2√2.

r = 4

h = 3 + 1 + √(r² – b²) = 4 + √(16 -8) = 4 + √8 ≅ 6.83.

Quiz Question

Number 223 of a series

Prove that fractions such as 8/7 must devolve into a repeating decimal sequence.

Update and Answer

The answer I gave when I was taking a math class about 1968 was much like Mike’s (see the comments section). It goes like this:

If you divide one integer by another there is either a remainder, or there is not a remainder (zero remainder). The remainder is always going to be less than the divisor, so if you continue repeating the process you will eventually produce a prior remainder. At this point the process will begin to repeat, and the result will be an infinite, repeating sequence.

No fair invoking Kolmogorov complexity. KC stipulates that the result of a process cannot require more information to describe than would be involved in describing the process. Remember, a process for computing the value of π requires only a few words to describe, and the result is a series of digits with no infinitely repeating pattern.

The difference between my answer and Mike’s is mine does not restrict the process to decimal numbers.

Quiz Question

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.

Update and Solution

This has received several responses, so it’s time to post a solution. See the following diagram.

R is the radius of the large circles. 1 is the radius of the small circle.

8R² = (2 + 2R)² = 4 + 8R + 4R²

4R² – 8R – 4 = 0

R = 1 + √2

Mike was the first with the correct answer.

Quiz Question

Number 221 of a series

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.

Update and Solution

Imagine the right hand side is the hypotenuse of a triangle, then the height of the triangle is 42. The 12 square and the 27 square imply the base is 15/12 the height. The area of the big square is

42² + (42 × 15/12)² = 4520.25

Quiz Question

Number 214 of a series

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.

Update and Solution

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)

Quiz Question

Number 212 of a series

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.

Update and Solution

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