A simple maths problem?

Anything goes, but keep it seemly...

A simple maths problem?

Postby udosuk » Sat Mar 24, 2007 9:11 am

The semester lasts 21 weeks.
The class consists of 21 students.
Each week 5 of the students are rostered to form a duty group.
At the end of the semester, each student must have been on duty for exactly 5 weeks.
Also, each student must have been grouped with everyone else.

Can you work out the groupings?

(Hint: use the 21 consonants to represent the students.)
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Postby coloin » Sat Mar 24, 2007 3:34 pm

Ah.. I see
Well the first 5 weeks would be simple...but the rest.....might be complicated

1 bcdfg
2 bhjkl
3 bmnpq
4 brstv
5 bwxyz

C

:!:
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Postby udosuk » Sat Mar 24, 2007 4:46 pm

Think smaller...

The library has 7 staff members. Each day of the week (Monday to Sunday) 3 members are rostered to work. Try to work out a weekly schedule that each member works 3 days a week and is teamed up with every other colleague once a week...
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Postby ronk » Sat Mar 24, 2007 5:32 pm

Nice puzzle, but checking it might be tougher than figuring it out.:D

1 bcdfg
2 bhjkl
3 bmnpq
4 brstv
5 bwxyz
6 chmrw
7 cjnsx
8 ckpty
9 clqvz
10 dhnsx
11 djpty
12 dkqvz
13 dlmrw
14 fhpry
15 fjqsz
16 fkmtw
17 flnvx
18 ghqtx
19 gjmvy
20 gknrz
21 glpsw
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Postby udosuk » Sun Mar 25, 2007 3:35 am

ronk wrote:Nice puzzle, but checking it might be tougher than figuring it out.:D

You're right, that's why your answer isn't correct...:)

ronk wrote:1 bcdfg
2 bhjkl
3 bmnpq
4 brstv
5 bwxyz
6 chmrw
7 cjnsx
8 ckpty
9 clqvz
10 dhnsx
11 djpty
12 dkqvz
13 dlmrw
14 fhpry
15 fjqsz
16 fkmtw
17 flnvx
18 ghqtx
19 gjmvy
20 gknrz
21 glpsw

Since each student must be grouped 5 times, each time with 4 others, that means one can't be grouped with the same person twice...

Check your groups 7 & 10... n,s,x all appear twice... So something must be wrong (e.g. x & p haven't been grouped together at all)...:idea:

There is a systematic way to do this... Trying my library example might give one some idea...:idea:
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Postby JPF » Sun Mar 25, 2007 8:28 am

ronk wrote:Nice puzzle, but checking it might be tougher than figuring it out.:D

these pairs don't meet :
hv,hz,jr,jw,ks,kx,lt,ms,mx,mz,nt,nw,pv,px,pz,qr,qw,rx,tz,vw

JPF
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Postby ronk » Sun Mar 25, 2007 11:20 am

udosuk wrote:
ronk wrote:Nice puzzle, but checking it might be tougher than figuring it out.:D

You're right, that's why your answer isn't correct...:)

There is a systematic way to do this...

Did have a system, but got lost using it. Sorry for not doing even a little checking.:(

Still don't have an easy way to check that each student is rostered with every other student exactly once, I'm pretty sure I at least followed "the system" this time.:)

1 bcdfg
2 bhjkl
3 bmnpq
4 brstv
5 bwxyz
6 chmrw
7 cjnsx
8 ckpty
9 clqvz
10 dhntz
11 djpvw
12 dkqrx
13 dlmsy
14 fhpry
15 fjqsz
16 fkmtw
17 flnvx
18 ghqtx
19 gjmvy
20 gknrz
21 glpsw

[edit: replaced roster, but 2nd wrong too]
Last edited by ronk on Sun Mar 25, 2007 10:28 am, edited 2 times in total.
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Postby re'born » Sun Mar 25, 2007 11:38 am

I'm not sure if this is the same as the same as ronk's list (as I am labeling from a-u) but here is my 2nd attempt. I did come up with a pseudo-systematic way to do it (using the odd permutations in the symmetric group on 4 letters).


Abcde
Afghi
Ajklm
Anopq
Arstu
Bfjnr
Bgkos
Bhlpt
Bimqu
Cflou
Cikpr
Cgmnt
Chjqs
Dgjpu
Dfkqt
Dhmor
Dilns
Ehknu
Efmps
Eglqr
Eijot


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Postby udosuk » Sun Mar 25, 2007 11:53 am

ronk, unfortunately you didn't pay attention to my last reply...

A very obvious way to check if your answer is wrong is to check if no 2 students are repeated twice or more...

From your new answer, group 7 & 10 still have n,s,x stuck together twice... So it's obviously a wrong answer...

Whatever system you're using, I'm afraid it's not working...:idea:

And nobody would take my advice of working out the 7-staff 7-day schedule first...:(

PS: Just noticed rep'nA posted an attempt before I posted this... From first look it looks alright, but apparently you're using a different system from mine and need to check to confirm later...
Last edited by udosuk on Sun Mar 25, 2007 8:03 am, edited 1 time in total.
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Postby re'born » Sun Mar 25, 2007 12:02 pm

udosuk wrote:
And nobody would take my advice of working out the 7-staff 7-day schedule first...:(


I did, I did!!!


abc
ade
afg
bdf
beg
cdg
cef
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Postby udosuk » Sun Mar 25, 2007 12:06 pm

Correct! And the next level would be to arrange 13 letters in 13 groups of 4... I won't spend time to think up a practical example this time...:)
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Postby re'born » Sun Mar 25, 2007 12:28 pm

udosuk wrote:Correct! And the next level would be to arrange 13 letters in 13 groups of 4... I won't spend time to think up a practical example this time...:)



abcd
aefg
ahij
aklm
behk
bfil
bgjm
ceim
cfjk
cghl
dfhm
dgik
dejl


[Edit: Are you going to suggest we do 31 letters in 31 groups of 6 next and then 43 letters in 43 groups of 7 (or have I not discovered the pattern)?]
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Postby ronk » Sun Mar 25, 2007 2:17 pm

udosuk wrote:Whatever system you're using, I'm afraid it's not working...:idea:

And nobody would take my advice of working out the 7-staff 7-day schedule first...:(

I did the 7-day thingie, but for me it's too small to see the pattern.

OK, new system ... revised result.
1 bcdfg
2 bhjkl
3 bmnpq
4 brstv
5 bwxyz
6 chmrw
7 cjnsx
8 ckpty
9 clqvz
10 dhntz
11 djpvw
12 dkqrx
13 dlmsy
14 fhpvx
15 fjqry
16 fkmsz
17 flntw
18 ghqsy
19 gjmtz
20 gknvw
21 glprx


And in roster form. (Sorry, but "code" and "tiny font" at the same time doesn't seem to work. Considering my record on this puzzle, however, "tiny" probably not required anyway.:) )
| 1 2 3 4 5 | 6 7 8 9 |10 11 12 13 |14 15 16 17 |18 19 20 21 |
--+---------------+------------+------------+------------+------------+
b | X X X X X | | | | |
c | X | X X X X | | | |
d | X | | X X X X | | |
f | X | | | X X X X | |
g | X | | | | X X X X |
--+---------------+------------+------------+------------+------------+
h | X | X | X | X | X |
j | X | X | X | X | X |
k | X | X | X | X | X |
l | X | X | X | X | X |
--+---------------+------------+------------+------------+------------+
m | X | X | X | X | X |
n | X | X | X | X | X |
p | X | X | X | X | X |
q | X | X | X | X | X |
--+---------------+------------+------------+------------+------------+
r | X | X | X | X | X |
s | X | X | X | X | X |
t | X | X | X | X | X |
v | X | X | X | X | X |
--+---------------+------------+------------+------------+------------+
w | X | X | X | X | X |
x | X | X | X | X | X |
y | X | X | X | X | X |
z | X | X | X | X | X |
--+---------------+------------+------------+------------+------------+
Last edited by ronk on Sun Mar 25, 2007 12:12 pm, edited 1 time in total.
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Postby udosuk » Sun Mar 25, 2007 2:47 pm

rep'nA wrote:[Edit: Are you going to suggest we do 31 letters in 31 groups of 6 next and then 43 letters in 43 groups of 7 (or have I not discovered the pattern)?]

You have discovered the pattern, and (31,31,6) is easier than (21,21,5), but if you can somehow do (43,43,7) I think a mathematician somewhere on this planet is willing to give you a million Euro...:!:

I'm seeing this thread has the potential to lead to another interesting line of discussion...:)

ronk wrote:OK, new system ... revised result.

Tough luck...:( Check out groups 11 & 20, v & w are stuck together twice...

There is a subtle trick somewhere to adapt from the case of (13,13,4)... If it's that easy it wouldn't be much fun right?:) (rep'nA, please wait for roughly a day for ronk to figure it out himself...:?: )

ronk wrote:Sorry, "code" and "tiny font" at the same time doesn't work.

Here is a way to display that table properly in tiny font...:idea:

  | 1  2  3  4  5 | 6  7  8  9 |10 11 12 13 |14 15 16 17 |18 19 20 21 |
--+---------------+------------+------------+------------+------------+
b | X  X  X  X  X |            |            |            |            |
c | X             | X  X  X  X |            |            |            |
d | X             |            | X  X  X  X |            |            |
f | X             |            |            | X  X  X  X |            |
g | X             |            |            |            | X  X  X  X |
--+---------------+------------+------------+------------+------------+
h |    X          | X          | X          | X          | X          |
j |    X          |    X       |    X       |    X       |    X       |
k |    X          |       X    |       X    |       X    |       X    |
l |    X          |          X |          X |          X |          X |
--+---------------+------------+------------+------------+------------+
m |       X       | X          |          X |       X    |    X       |
n |       X       |    X       | X          |          X |       X    |
p |       X       |       X    |    X       | X          |          X |
q |       X       |          X |       X    |    X       | X          |
--+---------------+------------+------------+------------+------------+
r |          X    | X          |       X    |    X       |          X |
s |          X    |    X       |          X |       X    | X          |
t |          X    |       X    | X          |          X |    X       |
v |          X    |          X |    X       | X          |       X    |
--+---------------+------------+------------+------------+------------+
w |             X | X          |    X       |          X |       X    |
x |             X |    X       |       X    | X          |          X |
y |             X |       X    |          X |    X       | X          |
z |             X |          X | X          |       X    |    X       |
--+---------------+------------+------------+------------+------------+

I guess it might be a better challenge for you guys than this puzzle itself...:)
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Postby Smythe Dakota » Sun Mar 25, 2007 3:25 pm

udosuk wrote:Think smaller...

The library has 7 staff members. Each day of the week (Monday to Sunday) 3 members are rostered to work. Try to work out a weekly schedule that each member works 3 days a week and is teamed up with every other colleague once a week...

This looks like projective geometry.:)

The axioms for a projective plane are:

A. On any two distinct points there is exactly one line.
B. On any two distinct lines there is exactly one point.
C. On each line there are at least 3 points.
D. Not all points are on the same line.

(Note: For point P to be on line L means the same thing as for line L to be on point P.)

If there are N points on each line, then there are N lines on each point, and N^2 - N + 1 points (and this same number of lines) altogether.

If N=3, one can construct a projective plane as follows. The points are: (1) the three vertices of an equilateral triangle, (2) the mid-points of the three sides, and (3) the center of the triangle. The lines are: (1) the three sides, (2) the three medians (joining the mid-point of each side to the opposite vertex), and (3) the inscribed circle.

Now just let each point be a library employee, and each line be a day of the week (or vice versa), and you have your solution.

Bill Smythe
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