BrainDen.com - Brain Teasers

Recommended Posts

Magic Belt - Back to the Cool Math Games

A magic wish-granting rectangular belt always shrinks to 1/2 its length and 1/3 its width whenever its owner makes a wish. After three wishes, the surface area of the belt’s front side was 4 cm2.

What was the original length, if the original width was 9 cm?

This old topic is locked since it was answered many times. You can check solution in the Spoiler below.

Pls visit New Puzzles section to see always fresh brain teasers.

Belt - solution

The original length of belt was 96 cm.

A magic rectangular belt always shrinks its length to 1/2 and width to 1/3 whenever its owner wishes something. After three such wishes, its surface (Edit: surface area of the front side) was 4 cm2. What was the original length, if the original width was 9 cm?

Share on other sites

• 2 months later... I am new here and I may be all wrong, but when I worked this out I got 192 or double 96.

Here's my work:

Original width = 9

1st wish: 9 * 1/3 = 3

2nd wish: 3 * 1/3 = 1

3rd wish: 1 * 1/3 = 1/3

Current width is 1/3 cm. Total area must be 1/3 * L = 16 cm

L = 16 / 1/3 = 48

Current Length must be 48 cm.

Working backwards: 3rd wish = 48

2nd wish = 48 * 2 = 96

1st wish = 96 * 2 = 192 original length.

What did I miss?

Share on other sites I am new here and I may be all wrong, but when I worked this out I got 192 or double 96.

Here's my work:

Original width = 9

1st wish: 9 * 1/3 = 3

2nd wish: 3 * 1/3 = 1

3rd wish: 1 * 1/3 = 1/3

Current width is 1/3 cm. Total area must be 1/3 * L = 16 cm

L = 16 / 1/3 = 48

Current Length must be 48 cm.

Working backwards: 3rd wish = 48

2nd wish = 48 * 2 = 96

1st wish = 96 * 2 = 192 original length.

What did I miss?

u got the area wrong

u said

Total area must be 1/3 * L = 16 cm

Its not 4 squared which does equal 16, it is 4 cm squared

ex: a square with a width and length of 2 = 4 cm2 not 16

so actually the total area of belt after 3 wishes is 1/3*L = 4 cm2

1/3= 0.33

4 * 0.33 = 12

12*2= 24

24*2= 48

48*2= 96

[if u really want to get technical 1/3 = 0.333333333333333333333333333333 and so on so its not exactly 96]

Share on other sites Oh - I get it - Thanks!

Share on other sites so actually the total area of belt after 3 wishes is 1/3*L = 4 cm2

1/3= 0.33

4 * 0.33 = 12

12*2= 24

24*2= 48

48*2= 96

[if u really want to get technical 1/3 = 0.333333333333333333333333333333 and so on so its not exactly 96]

1/3 does equal .33 repeated but your math is wrong.

4 * 0.33 does not = 12, but it doesn't matter because you were supposed to divide which does give you 12.

0.33 * L = 4 => L = 4 / 0.33

4 / 0.33 = 4 * 3 = 12, no decimals.

So the answer is exactly 96.

Share on other sites If you want to do it with the fraction instead of breaking it down to decimals it is even simpler.

4/(1/3)=4*3=12

Share on other sites Width after three wishes:

9*(1/3)*(1/3)*(1/3) = 9*(1/3)^3 = 9*(1/27) = 1/3

Length after three wishes:

surface = length * width

length = surface / width

length = 4/(1/3) = 4*3 = 12

Original length:

12/(1/2)/(1/2)/(1/2) = 12/(1/2)^3 = 12/(1/8) = 96

4 / (9 * (1/3)^3) / (1/2)^3 = 96 cm

4 / (9/27) * 8 = 96 cm

And the correct denotation of 1/3 would be 0.33... (with two or three dots) in which case the dots mean that the number is being repeated.

Share on other sites

• 2 weeks later... One other method of solving it would be,

If x,y are the original length and width resp.

xy=area

After each wish, length and width become 1/2 and 1/3 of the original resp.

Thus, the new area, after a wish = 1/6*old area.

As three wishes were made, and the final area being 4, the initial area is thus = 4*6*6*6 = 864

Width = 9 (given)

Length = 864/9=92.

Share on other sites

• 4 weeks later... this s not a puzzle at all....

Share on other sites

• 2 months later... let's assume original length is a and original width is b :

after 1st wish the area = a/2* b/3

after 2nd wish the area = a/4*b/9

after 3rd wish the area = a/8*b/27

the area after 3rd wish is given as 4 and the width is 9

therefore a/8*9/27=4 , 9a = 4*8*27

thus a = 96 cms

I hope it's the right way

Share on other sites

• 2 weeks later... 96 if I got my halvsies and thirdsies right.

by the by

heres the question..............

If a person with a 96cm waist was wearing the belt, would he/she be suffocated on the first, second, or third wish?

I would have wished the belt didn't shrink

Share on other sites Width is (((9/3)/3)/3) = 1/3 after 3 wishes

Area is 4cm (squared) so final length is 4 / 1/3 = 12

So original length is (((12*2)*2)*2) = 96

I know everyone has said it before, just confirming my working!

Share on other sites i tought he wished to make it a diffrent size....but if you want to go though all of that complication

Share on other sites

• 3 weeks later... Belt - Back to the Logic Puzzles

A magic rectangular belt always shrinks its length to 1/2 and width to 1/3 whenever its owner wishes something. After three such wishes, its surface was 4 cm2. What was the original length, if the original width was 9 cm?

I hate to be anal, but a belt is three dimensional. Even if the thickness of the belt were small enough to be neglegable, the belt would still have a front and a back. At most, the surface area of one side is 2 cm^2, resulting in an original length of 48 cm. You could just change the puzzle to state that "After three such wishes, the surface area of the front side was 4 cm^2."

Share on other sites

A magic rectangular belt always shrinks its length to 1/2 and width to 1/3 whenever its owner wishes something. After three such wishes, its surface was 4 cm2. What was the original length, if the original width was 9 cm?

I hate to be anal, but a belt is three dimensional. Even if the thickness of the belt were small enough to be neglegable, the belt would still have a front and a back. At most, the surface area of one side is 2 cm^2, resulting in an original length of 48 cm. You could just change the puzzle to state that "After three such wishes, the surface area of the front side was 4 cm^2."

puzzle edited

Share on other sites A magic rectangular belt always shrinks its length to 1/2 and width to 1/3 whenever its owner wishes something. After three such wishes, its surface was 4 cm2. What was the original length, if the original width was 9 cm?

I hate to be anal, but a belt is three dimensional. Even if the thickness of the belt were small enough to be neglegable, the belt would still have a front and a back. At most, the surface area of one side is 2 cm^2, resulting in an original length of 48 cm. You could just change the puzzle to state that "After three such wishes, the surface area of the front side was 4 cm^2."

puzzle edited

I believe this is one of those puzzles that you take literally and when the owner makes a wish the belts lengt goes to 1/2. Not that it is half each wish. It never says that it is each time.

Share on other sites Hello!

i am new to this.

Let me put the solution in this way.

Let the original lenght = x cm.

and the given width = 9 cm.

1st wish:

lenght =x/2 cm; width =9/3=3 cm;

2nd wish:

lenght =x/4 cm; width =1 cm;

3rd wish:

lenght =x/8 cm; width =1/3 cm;

After three wishes:

The given Surface area A (say) = 4 cm2;

Surface area A = length * width;

4 cm2 =(x/8 )*(1/3) cm2;

x = 4 *24= 96 cm;

The original length x= 96 cm;

Share on other sites I did it kinda like everyone else...

1. I divided 9 ( the original width) by 3, 3(new width) by 3, then 1( new, new , width) by 3, for the 3 wishes. This gave me 1/3 for the resulting width.

2. Next, i divided 4cm squared by 1/3, and got 12.

3. I doubled thise 3 times ( for the 3 wishes) to get 96, the original width.

Share on other sites

• 3 months later... simple i just used an equation to solve:

L= Length

W= Width

1/8L (1/27W) = 4

then just put 9 in for the width and solved for the length

(FYI i got 1/8 from placing 1/2 to the third power since it was wished to shrink 3 times, and did the same for the length- 1/3 ^ 3)

Share on other sites 96 cm

Edited by scuttill
Share on other sites

• 2 months later... l/8*w/27=4

l=4*8*27/9=96

Share on other sites

• 1 year later... I am new here and I may be all wrong, but when I worked this out I got 192 or double 96.

Here's my work:

Original width = 9

1st wish: 9 * 1/3 = 3

2nd wish: 3 * 1/3 = 1

3rd wish: 1 * 1/3 = 1/3

Current width is 1/3 cm. Total area must be 1/3 * L = 16 cm

L = 16 / 1/3 = 48

Current Length must be 48 cm.

Working backwards: 3rd wish = 48

2nd wish = 48 * 2 = 96

1st wish = 96 * 2 = 192 original length.

What did I miss?

you got the first bit right, the width after three wishes is 1/3cm, but then you said the area was 16cm. its meant to be 4cm^2, which is not the same as 16cm

so instead of 1/3 * L = 16cm, it should be

1/3 * L = 4 to find length after 3 wishes

i havent checked my answer, but looking at your post, that jumped out at me.

hope it helps

Share on other sites This topic is now closed to further replies.