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## #1 2006-01-05 04:45:39

Math Student
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### A Miscellaneous Problem that I can't work out!

24 unit cubes can be stuck together to make cuboids of different shapes. How many DIFFERENT cuboids can be made?

How many different cuboids can be made with:
a) 56 cubes
b) 100 cubes

## #2 2006-01-05 04:48:00

Math Student
Guest

### Re: A Miscellaneous Problem that I can't work out!

In fact, there is another question:

Many small cubes of side 1.2 are stuck together to make a large cube of 216cm³.
How many cubes are needed.

I worked it out to be 125 but can anyone confirm that answer?

## #3 2006-01-05 07:01:53

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

1x1x24
1x2x12
1x3x8
1x4x6
2x2x6
2x3x4
6 combinations.

IPBLE:  Increasing Performance By Lowering Expectations.

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## #4 2006-01-05 07:02:53

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

56=7x8

1x1x56
1x7x8

2 combinations

IPBLE:  Increasing Performance By Lowering Expectations.

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## #5 2006-01-05 07:04:29

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

100=5x5x4

1x1x100
1x4x25
1x5x20
4x5x5

4 combinations

IPBLE:  Increasing Performance By Lowering Expectations.

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## #6 2006-01-05 07:08:55

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

Let there be x cubes. the volume of each is 1.2^3=1.728 =>
x=216/1.728
x=125 cubes.

IPBLE:  Increasing Performance By Lowering Expectations.

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## #7 2006-01-05 07:14:22

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

you're absolutely right!

IPBLE:  Increasing Performance By Lowering Expectations.

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## #8 2006-01-05 07:26:13

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

### Re: A Miscellaneous Problem that I can't work out!

There are a few more combinations for 56 and 100, because 8 and 4 can be broken down more.

56 = 2x2x2x7

1x1x56
1x2x28
1x4x14
1x7x8
2x2x28
2x4x7

6 combinations.

100 = 2x2x5x5

1x1x100
1x2x50
1x4x25
1x5x20
1x10x10
2x2x25
2x5x10
4x5x5

8 combinations.

Why did the vector cross the road?
It wanted to be normal.

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## #9 2006-01-05 07:37:17

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

you're absolutey right!

IPBLE:  Increasing Performance By Lowering Expectations.

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## #10 2006-01-05 07:38:31

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

Can we find formula for this?

IPBLE:  Increasing Performance By Lowering Expectations.

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## #11 2006-01-05 07:39:33

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

If n is prime we have only 1 combination:
1x1xn

IPBLE:  Increasing Performance By Lowering Expectations.

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## #12 2006-01-05 07:41:24

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

If n is product of two prime we have 2 combinations:
1x1xn and
1xp1xp2

IPBLE:  Increasing Performance By Lowering Expectations.

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## #13 2006-01-05 07:42:45

Math Student
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### Re: A Miscellaneous Problem that I can't work out!

Wow! Thank you for your help!

## #14 2006-01-05 07:43:44

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

I thought out an algoritm that can be used

IPBLE:  Increasing Performance By Lowering Expectations.

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## #15 2006-01-05 08:03:21

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

### Re: A Miscellaneous Problem that I can't work out!

Be careful, though. As we can see from the example above, how many prime factors a number has isn't the only thing it depends on.

56 and 100 both have 4 prime factors, but they give answers of 6 and 8.

Why did the vector cross the road?
It wanted to be normal.

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## #16 2006-01-05 08:12:12

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

Yes, I saw that. My first result:

Ler dq[x] gives the number of the different divisors of x. Then the number of diffrerent rectangles that can be made with n squares is
Ceiling[dq[x]/2]

Last edited by krassi_holmz (2006-01-05 08:16:23)

IPBLE:  Increasing Performance By Lowering Expectations.

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## #17 2006-01-05 08:15:16

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

Example:
number:36
divisors:1,2,3,4,6,8,9,12,18,36
dq[36]=10

So we must have 5 rectangles. Here are they:
1x36
2x18
3x12
4x9
6x6

IPBLE:  Increasing Performance By Lowering Expectations.

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## #18 2006-01-05 08:27:14

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

Here is it in Mathematica language:
dq[x_] := Ceiling[Length[Divisors[x]]/2]

IPBLE:  Increasing Performance By Lowering Expectations.

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## #19 2006-01-05 08:32:34

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

### Re: A Miscellaneous Problem that I can't work out!

And here's a plot

Last edited by krassi_holmz (2006-01-05 08:33:24)

IPBLE:  Increasing Performance By Lowering Expectations.

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