Barry-Gordon

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

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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

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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

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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

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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 more…] about Stress Man­age­ment Test

Brain Health Newsletter, March Edition

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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 more…] about Brain Health Newslet­ter, March Edition

Brain Workout for Your Frontal Lobes

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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 more…] about Brain Work­out for Your Frontal Lobes