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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)

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).


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:

Mental Imagery and Spatial Rotation Brain Teaser

Here’s a fun puzzle that a friend gave me over dinner a few days ago …

How do you cut a cake into eight equal pieces with only three cuts?
the cake in the puzzle is not necessarily the one pictured below

mental rotation task

You have to use your mental rotation and mental imagery skills to visualize the answer for this puzzle. In doing so, you are using your visual cortex in the occipital lobes, your somatosensory cortex in your parietal lobes, and your executive functions in your frontal lobes to help create and evaluate your hypotheses.

Answer: Use two cuts to cut the cake into four equal pieces. Stack the four pieces vertically, and use your third cut to cut the four pieces in half horizontally.

Next brain teaser:

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)

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.

#14 Brain Teaser: Party For Polyglots

We are delighted to introduce you to Wes Carroll who has graciously created a few new puzzles to bend all those sharp brains out there! Wes Carroll

Wes is the head of Do The Math private tutoring services, Puzzle Master for the Ask A Scientist lecture series, and an internationally touring performer and teacher of music. With no further ado, the first puzzle!

Party For Polyglots

Difficulty: MEDIUM

Of the 100 people at a recent party, 90 spoke Spanish, 80 spoke Italian, and 75 spoke Mandarin. At least how many spoke all three languages?

Have you solved it yet? If you are working the problem, making hypotheses, testing your ideas, and coming up with a solution, you are using your frontal lobes. This is great exercise because the frontal lobes follow the “last hired, first fired” adage. They are they last areas of your brain to develop and the first to suffer the ravages of time and stress. So, keep exercising them!


10 could not speak Spanish, 20 could not speak Italian, and 25 could not speak Mandarin. So there could have been 10 people who spoke none of those languages.

However, that would maximize the number of people who could speak all three, and the problem asks at least how many speak all three. Therefore, we must assume that these 10, 20, and 25 people are all separate people. Having identified 55 each of whom is missing one language, the remaining 45 speak all three.

Next brain teaser in SharpBrains’ top 25 series:

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 »

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