This is from the '2008 MATHS CHALLENGE STAGE MATHEMATICS FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

I1. Copycats

Michelle's photocopier has seven buttons enabling her to reduce or
enlarge the area of a printed image. The area of the copy as a
percentage of the area of the original is specified on each button:
50% 75% 80% 100% 120% 125% 150%

c Using just four of its seven buttons and making copies of copies if
necessary, Michelle's photocopier can produce the same size images as
the complete set of seven buttons. Find the two sets of four buttons
with this property.

DO NOT GIVE ME ANY WORKING OUT!!!!!!!!!!!!!!!!
The first set is 50%, 80%, 120% 125%

YES OR NO, YES OR NO, YES OR NO?
IS ANOTHER SET EVEN POSSSSSSIBLE?

On Sat, 26 Apr 2008 01:34:09 -0700 (PDT), Albert
albert.xtheunknown0@gmail.com> wrote:

This is from the '2008 MATHS CHALLENGE STAGE MATHEMATICS FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

I1. Copycats

Michelle's photocopier has seven buttons enabling her to reduce or
enlarge the area of a printed image. The area of the copy as a
percentage of the area of the original is specified on each button:
50% 75% 80% 100% 120% 125% 150%

c Using just four of its seven buttons and making copies of copies if
necessary, Michelle's photocopier can produce the same size images as
the complete set of seven buttons. Find the two sets of four buttons
with this property.

DO NOT GIVE ME ANY WORKING OUT!!!!!!!!!!!!!!!!
The first set is 50%, 80%, 120% 125%

Yes.

quasi

On Sat, 26 Apr 2008 01:34:09 -0700 (PDT), Albert



albert.xtheunkno...@gmail.com> wrote:
This is from the '2008 MATHS CHALLENGE STAGE MATHEMATICS FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

I1. Copycats

Michelle's photocopier has seven buttons enabling her to reduce or
enlarge the area of a printed image. The area of the copy as a
percentage of the area of the original is specified on each button:
50% 75% 80% 100% 120% 125% 150%

c Using just four of its seven buttons and making copies of copies if
necessary, Michelle's photocopier can produce the same size images as
the complete set of seven buttons. Find the two sets of four buttons
with this property.

DO NOT GIVE ME ANY WORKING OUT!!!!!!!!!!!!!!!!
The first set is 50%, 80%, 120% 125%

YES OR NO, YES OR NO, YES OR NO?
IS ANOTHER SET EVEN POSSSSSSIBLE?

Yes.

quasi

On Sat, 26 Apr 2008 05:11:17 -0400, quasi <quasi@null.set> wrote:

On Sat, 26 Apr 2008 01:34:09 -0700 (PDT), Albert
albert.xtheunknown0@gmail.com> wrote:

This is from the '2008 MATHS CHALLENGE STAGE MATHEMATICS FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

I1. Copycats

Michelle's photocopier has seven buttons enabling her to reduce or
enlarge the area of a printed image. The area of the copy as a
percentage of the area of the original is specified on each button:
50% 75% 80% 100% 120% 125% 150%

c Using just four of its seven buttons and making copies of copies if
necessary, Michelle's photocopier can produce the same size images as
the complete set of seven buttons. Find the two sets of four buttons
with this property.

DO NOT GIVE ME ANY WORKING OUT!!!!!!!!!!!!!!!!
The first set is 50%, 80%, 120% 125%

To Albert:
No, that set doesn't work.

But there is at least one that does.

--Lynn

Yes. [There is another another one]

quasi

On Sat, 26 Apr 2008 18:57:51 GMT, "[Mr.] Lynn Kurtz"
kurtz@asu.edu.invalid> wrote:

On Sat, 26 Apr 2008 05:11:17 -0400, quasi <quasi@null.set> wrote:

On Sat, 26 Apr 2008 01:34:09 -0700 (PDT), Albert
albert.xtheunknown0@gmail.com> wrote:

This is from the '2008 MATHS CHALLENGE STAGE MATHEMATICS FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

I1. Copycats

Michelle's photocopier has seven buttons enabling her to reduce or
enlarge the area of a printed image. The area of the copy as a
percentage of the area of the original is specified on each button:
50% 75% 80% 100% 120% 125% 150%

c Using just four of its seven buttons and making copies of copies if
necessary, Michelle's photocopier can produce the same size images as
the complete set of seven buttons. Find the two sets of four buttons
with this property.

DO NOT GIVE ME ANY WORKING OUT!!!!!!!!!!!!!!!!
The first set is 50%, 80%, 120% 125%

To Albert:
No, that set doesn't work.

Actually, it does work.

On Sat, 26 Apr 2008 19:28:21 -0400, quasi <quasi@null.set> wrote:

On Sat, 26 Apr 2008 18:57:51 GMT, "[Mr.] Lynn Kurtz"
kurtz@asu.edu.invalid> wrote:

On Sat, 26 Apr 2008 05:11:17 -0400, quasi <quasi@null.set> wrote:

On Sat, 26 Apr 2008 01:34:09 -0700 (PDT), Albert
albert.xtheunknown0@gmail.com> wrote:

This is from the '2008 MATHS CHALLENGE STAGE MATHEMATICS FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

I1. Copycats

Michelle's photocopier has seven buttons enabling her to reduce or
enlarge the area of a printed image. The area of the copy as a
percentage of the area of the original is specified on each button:
50% 75% 80% 100% 120% 125% 150%

c Using just four of its seven buttons and making copies of copies if
necessary, Michelle's photocopier can produce the same size images as
the complete set of seven buttons. Find the two sets of four buttons
with this property.

DO NOT GIVE ME ANY WORKING OUT!!!!!!!!!!!!!!!!
The first set is 50%, 80%, 120% 125%

To Albert:
No, that set doesn't work.

Actually, it does work.

Perhaps I am reading in some restrictions that aren't there. If you
can only press each button once (?) those four give .6 magnification
while the full set gives .675, no? Are you allowing re-use of the
buttons, both in the original 7

and the subset of 4?

On Sun, 27 Apr 2008 00:01:54 GMT, "[Mr.] Lynn Kurtz"
kurtz@asu.edu.invalid> wrote:

On Sat, 26 Apr 2008 19:28:21 -0400, quasi <quasi@null.set> wrote:

On Sat, 26 Apr 2008 18:57:51 GMT, "[Mr.] Lynn Kurtz"
kurtz@asu.edu.invalid> wrote:

On Sat, 26 Apr 2008 05:11:17 -0400, quasi <quasi@null.set> wrote:

On Sat, 26 Apr 2008 01:34:09 -0700 (PDT), Albert
albert.xtheunknown0@gmail.com> wrote:

This is from the '2008 MATHS CHALLENGE STAGE MATHEMATICS FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

I1. Copycats

Michelle's photocopier has seven buttons enabling her to reduce or
enlarge the area of a printed image. The area of the copy as a
percentage of the area of the original is specified on each button:
50% 75% 80% 100% 120% 125% 150%

c Using just four of its seven buttons and making copies of copies if
necessary, Michelle's photocopier can produce the same size images as
the complete set of seven buttons. Find the two sets of four buttons
with this property.

DO NOT GIVE ME ANY WORKING OUT!!!!!!!!!!!!!!!!
The first set is 50%, 80%, 120% 125%

To Albert:
No, that set doesn't work.

Actually, it does work.

Perhaps I am reading in some restrictions that aren't there. If you
can only press each button once (?) those four give .6 magnification
while the full set gives .675, no? Are you allowing re-use of the
buttons, both in the original 7

You are only allowed to have 4 working buttons.

and the subset of 4?

Sure -- "copies of copies".

My interpretation of the problem is this ...

Find all sets of 4 buttons out of the original 7, such that, assuming
the other 3 are broken, the broken ones can nevertheless be emulated
by some sequence of the working ones.

Thinking real world, there's no reason to restrict a button to only
one use, and there was no such restriction mentioned in the statement
of the problem.

However, even if you did specify such a restriction, the OP's solution
is correct.

quasi

is there any systematic thought to find the second set?

Could you estimate (as a percentage) how much guessing-and-checking
there was in finding the second set?

On Apr 26, 7:11 pm, quasi <qu...@null.set> wrote:
On Sat, 26 Apr 2008 01:34:09 -0700 (PDT), Albert



albert.xtheunkno...@gmail.com> wrote:
This is from the '2008 MATHS CHALLENGE STAGE MATHEMATICS FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

I1. Copycats

Michelle's photocopier has seven buttons enabling her to reduce or
enlarge the area of a printed image. The area of the copy as a
percentage of the area of the original is specified on each button:
50% 75% 80% 100% 120% 125% 150%

c Using just four of its seven buttons and making copies of copies if
necessary, Michelle's photocopier can produce the same size images as
the complete set of seven buttons. Find the two sets of four buttons
with this property.

DO NOT GIVE ME ANY WORKING OUT!!!!!!!!!!!!!!!!
The first set is 50%, 80%, 120% 125%

YES OR NO, YES OR NO, YES OR NO?
IS ANOTHER SET EVEN POSSSSSSIBLE?

Yes.

quasi

Excluding using 100% is pointless, is there any systematic thought to
find the second set? Could you estimate (as a percentage) how much
guessing-and-checking there was in finding the second set?

On Sat, 26 Apr 2008 17:48:25 -0700 (PDT), Albert
albert.xtheunknown0@gmail.com> wrote:

On Apr 26, 7:11 pm, quasi <qu...@null.set> wrote:
On Sat, 26 Apr 2008 01:34:09 -0700 (PDT), Albert



albert.xtheunkno...@gmail.com> wrote:
This is from the '2008 MATHS CHALLENGE STAGE MATHEMATICS FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

I1. Copycats

Michelle's photocopier has seven buttons enabling her to reduce or
enlarge the area of a printed image. The area of the copy as a
percentage of the area of the original is specified on each button:
50% 75% 80% 100% 120% 125% 150%

c Using just four of its seven buttons and making copies of copies if
necessary, Michelle's photocopier can produce the same size images as
the complete set of seven buttons. Find the two sets of four buttons
with this property.

DO NOT GIVE ME ANY WORKING OUT!!!!!!!!!!!!!!!!
The first set is 50%, 80%, 120% 125%

YES OR NO, YES OR NO, YES OR NO?
IS ANOTHER SET EVEN POSSSSSSIBLE?

Yes.

quasi

Excluding using 100% is pointless, is there any systematic thought to
find the second set? Could you estimate (as a percentage) how much
guessing-and-checking there was in finding the second set?

Well, I saw the second set pretty much right away.

Firstly, use _fractions_, rather than percents, so you can see the
true numerators and denominators.

Thus, you have the buttons

1/2, 3/4, 4/5, 1, 6/5, 5/4, 3/2

Copies of copies means you can multiply fractions (for the buttons
that you select).

My heuristic reasoning was as follows.

The button 1 seems wasted, as long as it's recoverable -- so forget
that one.

You must use 5/4, otherwise there's no way to get a numerator which is
a multiple of 5.

Similarly you must use either 4/5 or 6/4

(or both), otherwise you
can't get a denominator which is a multiple of 5.

The button 1/2 seems hard to get unless you already have it, so I
tentatively selected it.

I can see that (1/2)*(3/2) = 3/4, so 3/2 was my next selection.

To get 1, why not take the pair 4/5 and 5/4.

Thus, as tentative selections, I now have 1/2, 4/5, 5/4, 3/2.

I already know I can generate 3/4 and 1, so the only remaining button
to emulate is 6/5.

But (4/5)*(3/2) = 6/5, so the solution works.

Now that was heuristic.

As far as a more rigorous technique, let me give it a shot.

A natural approach is to use prime factorization ...

Thus, write each fraction as a prime power or a product of prime
powers. The only relevant primes are those which are factors of either
the numerator or the denominator, namely 2, 3, 5.

1/2 = 2^(-1) * 3^(0) * 5^(0) <--> [-1,0,0]
3/4 = 2^(-2) * 3^(1) * 5^(0) <--> [-2,1,0]
4/5 = 2^(2) * 3^(0) * 5^(-1) <--> [2,0,-1]
1 = 2^(0) * 3^(0) * 5^(0) <--> [0,0,0]
6/5 = 2^(1) * 3^(1) * 5^(-1) <--> [1,1,-1]
5/4 = 2^(-2) * 3^(0) * 5^(1) <--> [-2,0,1]
3/2 = 2^(-1) * 3^(1) * 5^(0) <--> [-1,1,0]

Thus, we can recast the problem as follows ...

Find all sets of 4 vectors from the 7 vectors

v_1 = [-1,0,0]
v_2 = [-2,1,0]
v_3 = [2,0,-1]
v_4 = [0,0,0]
v_5 = [1,1,-1]
v_6 = [-2,0,1]
v_7 = [-1,1,0]

such that, using nonnegative integer linear combinations, you can
generate the other 3.

Only one vector, v_6, has a positive 3rd components, so we need v_6.

Only one vector, v_3, has a negative sum for the 2nd and 3rd
components, so we need v_3.

Since v_3 + v_6 = v_4, we don't need v_4.

So far, we have 2 vectors already selected for the team, namely

v_3, v_6

so we only need 2 more vectors, but not v_4.

Thus, of the remaining 4 candidates, v_1, v_2, v_5, v_7, we need two
of them.

Maybe there's something simple that can narrow it down further, but at
this point, I don't see anything immediate except to try each of the 6
pairs, and see if the result works.

Thus, you have the buttons
1/2, 3/4, 4/5, 1, 6/5, 5/4, 3/2
Copies of copies means you can multiply fractions (for the buttons
that you select).
[...]
You must use 5/4, otherwise there's no way to get a numerator which is
a multiple of 5.

Thus, we can recast the problem as follows ...
Find all sets of 4 vectors from the 7 vectors
[...]
such that, using nonnegative integer linear combinations, you can
generate the other 3.