## #22 Brain Teaser: The Really, Really, Really Big Number

Here is new brain teaser written by puzzle master Wes Carroll.

The Really, Really, Really Big Number

Difficulty: HARD
Type: MATH (Numerical/Abstract)
Intimidation Factor: HIGH — but don’t be scared!

Question:
When you divide 12 by 5, the remainder is 2; it’s what’s left over after you have removed all the 5s from the 12.  When you raise 4 to the fifth power (that is, 45), you multiply four by itself five times: 4x4x4x4x4, which equals 1,024.

What is the remainder when you divide 100100 by 11?

Solution:
1

Explanation:
This one is so sneaky.

First, consider 100 divided by 11. The remainder here is 1. Now consider the remainder when 100×100 is divided by 11. Don’t do it on your calculator or on paper. Rather, consider that you have one hundred hundreds, and each of them has a remainder of 1 when divided by 11. So, go through each of your hundred hundreds and divide it by 11, leaving remainder 1. Then collect up your remainders into a single hundred, and divide it by 11, leaving a remainder of 1. This process can be extended to dividing 100x100x100 by 11, and indeed, to dividing any power of 100 by 11.

Next brain teaser in SharpBrains’ top 25 series:

## Math Brain Teaser: Concentric Shapes or The Unkindest Cut of All, Part 2 of 2

If you missed Part 1, also written by puzzle master Wes Carroll, you can start there and then come back here to Part 2.

Concentric Shapes:
The Unkindest Cut of All, Part 2 of 2

Difficulty: HARDER
Type: MATH (Spatial)

Question:
Imagine a square within a circle within a square. The circle just grazes each square at exactly four points. Find the ratio of the area of the larger square to the smaller.

In this puzzle you are working out many of the same skills as in Part I: spatial visualization (occipital lobes), memory (temporal lobes), logic (frontal lobes), planning (frontal lobes), and hypothesis generation (frontal lobes).

Solution:
Two to one.

Explanation:
Draw the smaller square’s diagonal to see that the the smaller square’s diagonal is the diameter of the circle. Divide the larger square into two equal rectangular halves to see that the larger square’s side is also the diameter of the circle. This means that the smaller square’s diagonal equals the larger square’s side. (Or, if you prefer, simply rotate the inner square by 45 degrees.) As we’ve seen in the earlier puzzle “The Unkindest Cut Of All,” the area of the smaller square is half that of the larger, making the ratio two to one.

More brain teaser games:

## Math Brain Teaser: The Unkindest Cut of All, Part 1 of 2

In honor of Mathematics Awareness Month, here is another mathematical brain bender from puzzle master Wes Carroll.

The Unkindest Cut of All, Part 1 of 2

Difficulty: HARD
Type: MATH (Spatial)

Question:
The area of a square is equal to the square of the length of one side. So, for example, a square with side length 3 has area (32), or 9. What is the area of a square whose diagonal is length 5?

In this puzzle you are working out your spatial visualization (occipital lobes), memory (temporal lobes), and hypothesis generation (frontal lobes).

Solution:
12.5

Explanation:
I am especially fond of these two ways to solve this problem:

1. Draw the right triangle whose hypotenuse is the square’s diagonal, and whose two legs are two sides of the square. Then use the Pythagorean Theorem (a^2 + b^2 = c^2) to solve for the length of each side. Since two sides are equal, we get (a^2 + a^2 = c^2), or (2(a^2) = c^2) ). Since c is 5, 2(a^2) = 25, making a^2 equal to 25/2, or 12.5. Since the area of the square is a^2, we’re done: it’s 12.5.

2. Tilt the square 45 degrees and draw a square around it such the the corners of the original square just touch the middles of the sides of the new, larger square. The new square has sides each 5 units long (the diagonal of the smaller square), and it therefore has area 25. However, a closer inspection reveals that the area of the larger square must be exactly twice that of the smaller. Therefore the smaller square has area 25/2, or 12.5.

You can now go on to Concentric Shapes: The Unkindest Cut of All, Part 2 of 2

More brain teaser games:

## Brain Teaser: Dr. Nasty’s Giant Cube

Here is another mind-bender created by Wes Carroll for the SharpBrains readers.

Presenting …
Dr. Nasty’s Giant Cube

Difficulty: HARDER
Type: HYBRID (Logic/Spatial)

Question:
The diabolical Dr. Nasty has turned his Growth Ray on a perfect cube that used to measure one foot on a side. The new larger cube has twice the surface area of the original. Find the volume of the larger cube.

Click to read the Solution and Explanation.

## Stress Management Test

Here’s a quick test to determine your stress level.  Read the following description completely before looking at the picture.

The picture below was used in a case study on stress levels at St. Mary’s Hospital. Look at both dolphins jumping out of the water. The dolphins are identical. A closely monitored, scientific study revealed that, in spite of the fact that the dolphins are identical, a person under stress would find differences between the two dolphins. The more differences a person finds between the dolphins, the more stress that person is experiencing.

Look at the photograph, and if you find more than one or two differences, you may want to take a vacation or at least get a massage.

As seen in The New York Times, The Wall Street Journal, CNN, and more, SharpBrains is an independent market research firm tracking applied brain science. Explore our most popular resources HERE.

### Top 10 Brain Teasers and Illusions

1. You think you know the colors? Try the Stroop Test
2. Check out this brief attention experiment