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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)
Vitruvian Man

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)
Square

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.

cube brain teaser

Click to read Hint #1.

Click to read Hint #2.

Click to read Hint #3.

Click to read Hint #4.

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.

–> CLICK HERE to see the picture before reading more.

Read the rest of this entry »

Brain Health Newsletter, March Edition

We hope you are enjoying Brain Awareness Week this week and hopefully thinking a little more about your brain and brain fitness! Below you have the Brain Fitness Newsletter we sent a few days ago. You can subscribe to this monthly email update in the box on the the top of this page.

We have had another busy month behind us, and we’re looking forward to Brain Awareness Week March 12-18. Keep reading for the details (including a special offer in honor of Brain Awareness Week) …

I. Press Coverage
II. Events
III. Program Reviews
IV. New Offerings
V. Website and Blog Summary, including brain teasers

Read the rest of this entry »

Brain Workout for Your Frontal Lobes

Your frontal lobes are home to your executive functions, including pattern recognition. Here’s a puzzle to challenge your ability to uncover a pattern.

In this puzzle, three numbers: 16, 14, and 38, need to be assigned to one of the rows of numbers below. To which row should each number be assigned – A, B, or C?

A: 0 6 8 9 3
B: 5 13 2 10 16
C: 7 1 47 11 17

Why do we care about pattern recognition skills? Well, if you’re an athlete, then you want to constantly improve your ability to see spatial patterns on the court or field quickly so you can act on them – by passing to open space or attacking the goal at the right moment. Stock traders look for patterns in the market behavior to guide them on buying and selling decisions. Chess masters are experts at recognizing complicated moves. Reading is also pattern recognition.

So, you use pattern recognition all the time whether you know it or not. But remember, using a skill is great, but you have to keep exercising it a little bit harder each time to develop it further.

Have you solved the puzzle yet? If not, here’s a hint:
It’s not a mathematical problem. The numerical values are irrelevant.

Keep reading for the answer
Read the rest of this entry »

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