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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.

 

PS: Enjoy these 50 brain teasers to test your cog­ni­tive abil­ity. Free, and fun for adults of any age!

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

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.

Brain Teaser for Stress

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 »

Exercise Your Brains — Visual Logic Brain Teaser

In which direc­tion is the bus pic­tured below traveling?

school bus attention game

Do you know the answer?

The only pos­si­ble answers are “left” or “right.”

Still don’t know?

Keep read­ing for the answer and explanation…

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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