The joy of
scheduling chores
Dr. C
Introducing: The MathPath chores!
Introducing: The MathPath chores!
Seven students do chores for the week
Three chores per day
Seven days per week
Introducing: The MathPath chores!
Seven students do chores for the week
Three chores per day
Seven days per week
Everyone must work the same number of days
Every pair of students must work together
Tuesday:
Wednesday:
Thursday:
Friday:
Saturday:
Sunday:
Monday:
Who needs to work with
Tuesday:
Wednesday:
Thursday:
Friday:
Saturday:
Sunday:
Monday:
?
F
G
E
D
A
B
C
Geometry?!
F
G
E
D
A
B
Point ←→ Student
Line ←→ Day of chores
C
F
G
E
D
A
B
C
Chores: Every day, three students do chores.
F
G
E
D
A
B
C
Chores: Every day, three students do chores.
Geometry: Every line has three points.
F
G
E
D
A
B
C
Chores: Every 2 students work together one day.
F
G
E
D
A
B
C
Chores: Every 2 students work together one day.
Geometry: Every 2 points are together on one line.
F
G
E
D
A
B
C
Geometry: Every point has 3 lines through it.
F
G
E
D
A
B
C
Chores: Every student works on 3 days.
Geometry: Every point has 3 lines through it.
F
G
E
D
A
B
C
Geometry: Any 2 lines intersect in one point.
F
G
E
D
A
B
C
Chores: On any 2 days one student works on both.
Geometry: Any 2 lines intersect in one point.
F
G
E
D
A
B
A finite geometry
C
F
G
E
D
A
B
A projective geometry
C
F
G
E
D
A
B
The Fano Plane
C
F
G
E
D
A
B
(7,3,1)
C
This kind of geometry is called a Steiner system.
Ingredients:
Points
Lines
Rules the lines must follow:
Each line has the same size
Every 2 points appear in exactly one line
balance & fairness
Pop (or soda) quiz: Is this a Steiner system?
Points: {A, B, C, D, E, F , G, H, I}
Lines:
A−B−C
D−E −F
G−H −I
Pop quiz: Is this a Steiner system?
Points: {A, B, C, D}
Lines: All 3-point subsets:
A−B−C
A−B−D
A−C−D
B−C−D
Wait a minute. . .
When we built this schedule, we did not
force everyone to work the same number of days
F
Seven students
Three chores per day
Seven days per week
G
Everyone must work the same
number of days
Every pair of students must work
together
E
D
A
B
C
Wait a minute. . .
When we built this schedule, we did not
force everyone to work the same number of days
F
G
E
D
A
B
C
In a Steiner system with
v points and k points per line:
r lines containing P
v − 1 other points
P
This replication number is automatic in every
Steiner system!
r =
v −1
(students) − 1
=
(students per day) − 1
k −1
Puzzle! What if we had 8 students working?
Careful! r =
v −1
is not enough.
k −1
v = 5, k = 3:
Were we lucky to end up with exactly 7 days?
Were we lucky to end up with exactly 7 days?
There are
pairs of students.
And
pairs work together each day.
So we need
days.
In general, there are
(this number is also free!)
lines.
A short history of Steiner systems
Steiner systems encode balance and fairness:
1782: Leonhard Euler’s
“36 officers” problem:
Is it possible to arrange six
regiments consisting of six
officers each of different
ranks in a 6 × 6 square so
that no rank or regiment will
be repeated in any row or
column?
A short history of awesome middle names
Steiner systems encode balance and fairness:
1847: Rev. Thomas Penyngton
Kirkman’s “15 schoolgirls”
problem:
“Fifteen young ladies in a
school walk out in rows of
three for seven days in
succession; it is required
to arrange them daily so
that no two shall walk twice
in the same row.”
A short history of awesome middle names
Steiner systems encode balance and fairness:
1920s: Sir Ronald Aylmer Fisher
links Steiner systems to
statistical experiments:
Design an experiment in
which we test 7 fertilizers at
a set of farms (each with 3
plots) such that every pair
of fertilizers are tested
together at some farm.
A short history of awesome middle names
Steiner systems encode balance and fairness:
1940s: Claude Elwood Shannon
founds coding theory;
Richard Wesley
Hamming creates the
first practical codes:
Steiner systems
underlie them!
A short history of awesome middle names
Steiner systems encode balance and fairness:
1940s: Claude Elwood Shannon
founds coding theory;
Richard Wesley
Hamming creates the
first practical codes:
Steiner systems
underlie them!
A short history of Steiner systems
Steiner systems encode balance and fairness:
Now: Steiner systems solve
problems in statistics,
scheduling,
communications, and
anywhere else where
fairness and balance are
important. Even quantum
computers!
New rules: Class groups
9 students
3 groups of 3 students each
Everyone must work with everyone else
Switch groups each day.
New rules: Class groups
9 points
3 groups of 3 students each
Everyone must work with everyone else
Switch groups each day.
New rules: Class groups
9 points
3 parallel lines with 3 points each.
Everyone must work with everyone else
Switch groups each day.
New rules: Class groups
9 points
3 parallel lines with 3 points each.
Every 2 points have one line through them.
Switch groups each day.
New rules: Class groups
9 points
3 parallel lines with 3 points each.
Every 2 points have one line through them.
Switch groups each day.
How many groups will each student be in?
G
H
I
D
E
F
A
B
C
G
H
I
D
E
F
A
B
C
A finite geometry
G
H
I
D
E
F
A
B
C
An affine geometry
G
H
I
D
E
F
A
B
C
An affine plane
G
H
I
D
E
F
A
B
C
A Steiner system
Done!
The many names of (7,3,1) by Ezra Brown,
Mathematics Magazine April 2002
Many more names of (7,3,1) by Ezra Brown,
Mathematics Magazine April 2015