Alcuin’s Mathematical Puzzles

Annie Raymond

University of Washington

March 23, 2015

Alcuin’s problemAlcuin wrote Propositions ad acuendos iuvenes, i.e. problems to sharpenthe young, which contains the following problem:

A man had to transport to the far side of a river a wolf, a goat, and abundle of cabbages. The only boat he could find was one which couldcarry only two of them. For that reason, he sought a plan which wouldenable them all to get to the far side unhurt. Let him, who is able, sayhow it could be possible to transport them safely.

Solution:

The man first brings the goat to the far side and leaves it there.

He goes back and brings the wolf with him, leaving it on the far side,and brings back the goat.

He leaves the goat on the initial side, takes the cabbage and brings itto the far side.

Finally, he goes back to the original shore, and takes the goat to bringit to the far side.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 2 / 13

Alcuin’s problemAlcuin wrote Propositions ad acuendos iuvenes, i.e. problems to sharpenthe young, which contains the following problem:

A man had to transport to the far side of a river a wolf, a goat, and abundle of cabbages. The only boat he could find was one which couldcarry only two of them. For that reason, he sought a plan which wouldenable them all to get to the far side unhurt. Let him, who is able, sayhow it could be possible to transport them safely.

Solution:

The man first brings the goat to the far side and leaves it there.

He goes back and brings the wolf with him, leaving it on the far side,and brings back the goat.

He leaves the goat on the initial side, takes the cabbage and brings itto the far side.

Finally, he goes back to the original shore, and takes the goat to bringit to the far side.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 2 / 13

Alcuin’s problemAlcuin wrote Propositions ad acuendos iuvenes, i.e. problems to sharpenthe young, which contains the following problem:

A man had to transport to the far side of a river a wolf, a goat, and abundle of cabbages. The only boat he could find was one which couldcarry only two of them. For that reason, he sought a plan which wouldenable them all to get to the far side unhurt. Let him, who is able, sayhow it could be possible to transport them safely.

Solution:

The man first brings the goat to the far side and leaves it there.

He goes back and brings the wolf with him, leaving it on the far side,and brings back the goat.

He leaves the goat on the initial side, takes the cabbage and brings itto the far side.

Finally, he goes back to the original shore, and takes the goat to bringit to the far side.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 2 / 13

Alcuin’s problemAlcuin wrote Propositions ad acuendos iuvenes, i.e. problems to sharpenthe young, which contains the following problem:

A man had to transport to the far side of a river a wolf, a goat, and abundle of cabbages. The only boat he could find was one which couldcarry only two of them. For that reason, he sought a plan which wouldenable them all to get to the far side unhurt. Let him, who is able, sayhow it could be possible to transport them safely.

Solution:

The man first brings the goat to the far side and leaves it there.

He goes back and brings the wolf with him, leaving it on the far side,and brings back the goat.

He leaves the goat on the initial side, takes the cabbage and brings itto the far side.

Finally, he goes back to the original shore, and takes the goat to bringit to the far side.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 2 / 13

Alcuin’s problemAlcuin wrote Propositions ad acuendos iuvenes, i.e. problems to sharpenthe young, which contains the following problem:

A man had to transport to the far side of a river a wolf, a goat, and abundle of cabbages. The only boat he could find was one which couldcarry only two of them. For that reason, he sought a plan which wouldenable them all to get to the far side unhurt. Let him, who is able, sayhow it could be possible to transport them safely.

Solution:

The man first brings the goat to the far side and leaves it there.

He goes back and brings the wolf with him, leaving it on the far side,and brings back the goat.

He leaves the goat on the initial side, takes the cabbage and brings itto the far side.

Finally, he goes back to the original shore, and takes the goat to bringit to the far side.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 2 / 13

Alcuin’s problemAlcuin wrote Propositions ad acuendos iuvenes, i.e. problems to sharpenthe young, which contains the following problem:

A man had to transport to the far side of a river a wolf, a goat, and abundle of cabbages. The only boat he could find was one which couldcarry only two of them. For that reason, he sought a plan which wouldenable them all to get to the far side unhurt. Let him, who is able, sayhow it could be possible to transport them safely.

Solution:

The man first brings the goat to the far side and leaves it there.

He goes back and brings the wolf with him, leaving it on the far side,and brings back the goat.

He leaves the goat on the initial side, takes the cabbage and brings itto the far side.

Finally, he goes back to the original shore, and takes the goat to bringit to the far side.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 2 / 13

Alcuin’s modified problem

A man had to transport to the far side of a river a wolf, a goat, a rabbitand a bundle of cabbages. The only boat he could find was one whichcould carry only two of them. For that reason, he sought a plan whichwould enable them all to get to the far side unhurt. Let him, who is able,say how it could be possible to transport them safely.

Anyone?

Clearly impossible! Whatever the man takes first in the boat, there willremain an animal eating something else on the shore.

Let’s represent that mathematically!

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 3 / 13

Alcuin’s modified problem

A man had to transport to the far side of a river a wolf, a goat, a rabbitand a bundle of cabbages. The only boat he could find was one whichcould carry only two of them. For that reason, he sought a plan whichwould enable them all to get to the far side unhurt. Let him, who is able,say how it could be possible to transport them safely.

Anyone?

Clearly impossible! Whatever the man takes first in the boat, there willremain an animal eating something else on the shore.

Let’s represent that mathematically!

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 3 / 13

Alcuin’s modified problem

A man had to transport to the far side of a river a wolf, a goat, a rabbitand a bundle of cabbages. The only boat he could find was one whichcould carry only two of them. For that reason, he sought a plan whichwould enable them all to get to the far side unhurt. Let him, who is able,say how it could be possible to transport them safely.

Anyone?

Clearly impossible! Whatever the man takes first in the boat, there willremain an animal eating something else on the shore.

Let’s represent that mathematically!

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 3 / 13

Alcuin’s modified problem

A man had to transport to the far side of a river a wolf, a goat, a rabbitand a bundle of cabbages. The only boat he could find was one whichcould carry only two of them. For that reason, he sought a plan whichwould enable them all to get to the far side unhurt. Let him, who is able,say how it could be possible to transport them safely.

Anyone?

Clearly impossible! Whatever the man takes first in the boat, there willremain an animal eating something else on the shore.

Let’s represent that mathematically!

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 3 / 13

Graph representation

We make a graph where each vertex (point) represents ananimal/vegetable/person/thing, and we put an edge between two of themif they cannot be left alone without supervision. We call this graph theAlcuin graph of the problem.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 4 / 13

Graph representation

We make a graph where each vertex (point) represents ananimal/vegetable/person/thing, and we put an edge between two of themif they cannot be left alone without supervision. We call this graph theAlcuin graph of the problem.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 4 / 13

Graph representation

We make a graph where each vertex (point) represents ananimal/vegetable/person/thing, and we put an edge between two of themif they cannot be left alone without supervision. We call this graph theAlcuin graph of the problem.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 4 / 13

Graph representation

We make a graph where each vertex (point) represents ananimal/vegetable/person/thing, and we put an edge between two of themif they cannot be left alone without supervision. We call this graph theAlcuin graph of the problem.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 4 / 13

Graph representation

We make a graph where each vertex (point) represents ananimal/vegetable/person/thing, and we put an edge between two of themif they cannot be left alone without supervision. We call this graph theAlcuin graph of the problem.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 4 / 13

Solution to modified problem

We need a boat that has two places (plus one for the farmer).

Then we can either first take the wolf and the cabbage to the other sideand leave them there, and come back to the original side and get therabbit and goat or we can do it the other way around by taking first therabbit and goat, and then the wolf and cabbage.

In both cases, the first trip leaves a graph without any edges on theoriginal shore; i.e., the vertices we removed were adjacent to every edge.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 5 / 13

Solution to modified problem

We need a boat that has two places (plus one for the farmer).

Then we can either first take the wolf and the cabbage to the other sideand leave them there, and come back to the original side and get therabbit and goat or we can do it the other way around by taking first therabbit and goat, and then the wolf and cabbage.

In both cases, the first trip leaves a graph without any edges on theoriginal shore; i.e., the vertices we removed were adjacent to every edge.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 5 / 13

Solution to modified problem

We need a boat that has two places (plus one for the farmer).

Then we can either first take the wolf and the cabbage to the other sideand leave them there, and come back to the original side and get therabbit and goat or we can do it the other way around by taking first therabbit and goat, and then the wolf and cabbage.

In both cases, the first trip leaves a graph without any edges on theoriginal shore; i.e., the vertices we removed were adjacent to every edge.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 5 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Definition

A stable set in a graph is a set of vertices (points) for which there is not asingle edge (line) between them.

Definition

A vertex cover in a graph is a set of vertices (points) such that every edge(line) in the graph is adjacent to a vertex in that set.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 6 / 13

Stable set and vertex cover

Theorem

The size of the minimum vertex cover plus the size of the maximum stableset is equal to the number of vertices.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 7 / 13

Generalization

Proposition

If there exists a solution to an Alcuin-type problem, then the number ofextra places on the boat is at least the size of the minimum vertex coverof the Alcuin graph of the problem.

Proof.

If the boat has less places, then whatever elements we put on the boat forthe first trip will leave some edges in the Alcuin graph since thecorresponding elements cannot be a vertex cover.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 8 / 13

Generalization

Proposition

If there exists a solution to an Alcuin-type problem, then the number ofextra places on the boat is at least the size of the minimum vertex coverof the Alcuin graph of the problem.

Proof.

If the boat has less places, then whatever elements we put on the boat forthe first trip will leave some edges in the Alcuin graph since thecorresponding elements cannot be a vertex cover.

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 8 / 13

Minimum size of the boatWhat is the minimum boat size that will guarantee that there exists asolution?

wolf

goat

goose

cat

rabbitmouse

cabbage

carrot

grass

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 9 / 13

Minimum size of the boatWhat is the minimum boat size that will guarantee that there exists asolution?

Ben Affleck

Matt Damon

Jennifer Lopez

Angelina Jolie

George ClooneyJennifer Aniston

Brad Pitt

Billy Bob Thornton

Julia Roberts

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 9 / 13

Minimum size of the boatWhat is the minimum boat size that will guarantee that there exists asolution?

wolf

goat

goose

cat

rabbitmouse

cabbage

carrot

grass

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 9 / 13

Minimum size of the boatWhat is the minimum boat size that will guarantee that there exists asolution?

wolf

goat

goose

cat

rabbitmouse

cabbage

carrot

grass

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 9 / 13

Minimum size of the boat

What is the minimum boat size that will guarantee that there exists asolution?

wolf

goatgooserabbit mouse

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 10 / 13

Generalization 2

Proposition

If the boat has at least one extra place more than the size of the minimumvertex cover of the Alcuin graph of the problem, then there always exists asolution to the problem.

Proof.

Put all the elements corresponding to the minimum vertex cover of theproblem in the boat for all of the trips needed, and use the extra place tocarry all of the other elements one by one.

Surprisingly enough, it is hard to determine if a certain Alcuin-typeproblem needs to have at least a or a + 1 places in order for there to be asolution (where a is the size of the minimum vertex cover of the graph).

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 11 / 13

Generalization 2

Proposition

If the boat has at least one extra place more than the size of the minimumvertex cover of the Alcuin graph of the problem, then there always exists asolution to the problem.

Proof.

Put all the elements corresponding to the minimum vertex cover of theproblem in the boat for all of the trips needed, and use the extra place tocarry all of the other elements one by one.

Surprisingly enough, it is hard to determine if a certain Alcuin-typeproblem needs to have at least a or a + 1 places in order for there to be asolution (where a is the size of the minimum vertex cover of the graph).

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 11 / 13

Generalization 2

Proposition

If the boat has at least one extra place more than the size of the minimumvertex cover of the Alcuin graph of the problem, then there always exists asolution to the problem.

Proof.

Put all the elements corresponding to the minimum vertex cover of theproblem in the boat for all of the trips needed, and use the extra place tocarry all of the other elements one by one.

Surprisingly enough, it is hard to determine if a certain Alcuin-typeproblem needs to have at least a or a + 1 places in order for there to be asolution (where a is the size of the minimum vertex cover of the graph).

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 11 / 13

How do you find a minimum vertex cover or maximumstable set

Create a variable xv for every vertex v in the graph and let V be the set ofall the vertices.

Let E be set of all edges e = (i , j). Then the maximumsize of a stable set is equal to

max∑

v∈V xvsuch that xi + xj ≤ 1 for all e = (i , j) ∈ E

xv ∈ {0, 1} for all v ∈ V

The minimum size of a vertex cover is equal to

min∑

v∈V xvsuch that xi + xj ≥ 1 for all e = (i , j) ∈ E

xv ∈ {0, 1} for all v ∈ V

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 12 / 13

How do you find a minimum vertex cover or maximumstable set

Create a variable xv for every vertex v in the graph and let V be the set ofall the vertices. Let E be set of all edges e = (i , j).

Then the maximumsize of a stable set is equal to

max∑

v∈V xvsuch that xi + xj ≤ 1 for all e = (i , j) ∈ E

xv ∈ {0, 1} for all v ∈ V

The minimum size of a vertex cover is equal to

min∑

v∈V xvsuch that xi + xj ≥ 1 for all e = (i , j) ∈ E

xv ∈ {0, 1} for all v ∈ V

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 12 / 13

How do you find a minimum vertex cover or maximumstable set

Create a variable xv for every vertex v in the graph and let V be the set ofall the vertices. Let E be set of all edges e = (i , j). Then the maximumsize of a stable set is equal to

max∑

v∈V xvsuch that xi + xj ≤ 1 for all e = (i , j) ∈ E

xv ∈ {0, 1} for all v ∈ V

The minimum size of a vertex cover is equal to

min∑

v∈V xvsuch that xi + xj ≥ 1 for all e = (i , j) ∈ E

xv ∈ {0, 1} for all v ∈ V

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 12 / 13

How do you find a minimum vertex cover or maximumstable set

Create a variable xv for every vertex v in the graph and let V be the set ofall the vertices. Let E be set of all edges e = (i , j). Then the maximumsize of a stable set is equal to

max∑

v∈V xvsuch that xi + xj ≤ 1 for all e = (i , j) ∈ E

xv ∈ {0, 1} for all v ∈ V

The minimum size of a vertex cover is equal to

min∑

v∈V xvsuch that xi + xj ≥ 1 for all e = (i , j) ∈ E

xv ∈ {0, 1} for all v ∈ V

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 12 / 13

Thank you!

Annie Raymond (University of Washington) Alcuin’s Mathematical Puzzles March 23, 2015 13 / 13