… to this puzzle exists (but I didn’t see it).

Suppose you have an equal number of red and blue dots on a sheet of paper, and no three dots are co-linear.

Is it always possible to connect all the dots in red-blue pairs with straight lines, so that no two lines intersect?

Prove it.

The Rev Dodgson said:

… to this puzzle exists (but I didn’t see it).

Suppose you have an equal number of red and blue dots on a sheet of paper, and no three dots are co-linear.

Is it always possible to connect all the dots in red-blue pairs with straight lines, so that no two lines intersect?

Prove it.

nice pun, are you referring to the use of detergent solution bubbles to fit minimal surfaces to the problem

SCIENCE said:

The Rev Dodgson said:

… to this puzzle exists (but I didn’t see it).

Suppose you have an equal number of red and blue dots on a sheet of paper, and no three dots are co-linear.

Is it always possible to connect all the dots in red-blue pairs with straight lines, so that no two lines intersect?

Prove it.

nice pun, are you referring to the use of detergent solution bubbles to fit minimal surfaces to the problem

No pun intended :) (and the solution given did not involve detergent)

An example of the problem, in case it wasn’t clear:

The Rev Dodgson said:

… to this puzzle exists (but I didn’t see it).

Suppose you have an equal number of red and blue dots on a sheet of paper, and no three dots are co-linear.

Is it always possible to connect all the dots in red-blue pairs with straight lines, so that no two lines intersect?

Prove it.

Sydney Uni Maths Society used to (and maybe still do) have an annual maths competition, consisting of ten problems each, available to anyone enrolled at any Australian university. This very problem was on the first problem set I entered. It’s quite easy to prove with induction (the method I used,) or with reductio ad absurdum.

<spoiler>:
It is always possible.

The Rev Dodgson said:

SCIENCE said:

The Rev Dodgson said:

… to this puzzle exists (but I didn’t see it).

Suppose you have an equal number of red and blue dots on a sheet of paper, and no three dots are co-linear.

Is it always possible to connect all the dots in red-blue pairs with straight lines, so that no two lines intersect?

Prove it.

nice pun, are you referring to the use of detergent solution bubbles to fit minimal surfaces to the problem

No pun intended :) (and the solution given did not involve detergent)

An example of the problem, in case it wasn’t clear:

based on your diagram, we suppose you can argue yes without the use of soap

btm said:

The Rev Dodgson said:

… to this puzzle exists (but I didn’t see it).

Suppose you have an equal number of red and blue dots on a sheet of paper, and no three dots are co-linear.

Is it always possible to connect all the dots in red-blue pairs with straight lines, so that no two lines intersect?

Prove it.

Sydney Uni Maths Society used to (and maybe still do) have an annual maths competition, consisting of ten problems each, available to anyone enrolled at any Australian university. This very problem was on the first problem set I entered. It’s quite easy to prove with induction (the method I used,) or with reductio ad absurdum.

<spoiler>:
It is always possible.

Yes, the answer and the proof are quite easy (only very basic geometry required) – once you have seen it.

So did you get it right when you did the competition?

The Rev Dodgson said:

btm said:

The Rev Dodgson said:

… to this puzzle exists (but I didn’t see it).

Suppose you have an equal number of red and blue dots on a sheet of paper, and no three dots are co-linear.

Is it always possible to connect all the dots in red-blue pairs with straight lines, so that no two lines intersect?

Prove it.

Sydney Uni Maths Society used to (and maybe still do) have an annual maths competition, consisting of ten problems each, available to anyone enrolled at any Australian university. This very problem was on the first problem set I entered. It’s quite easy to prove with induction (the method I used,) or with reductio ad absurdum.

<spoiler>:
It is always possible.

Yes, the answer and the proof are quite easy (only very basic geometry required) – once you have seen it.

So did you get it right when you did the competition?

does topology count as geometry

SCIENCE said:

The Rev Dodgson said:

btm said:

Sydney Uni Maths Society used to (and maybe still do) have an annual maths competition, consisting of ten problems each, available to anyone enrolled at any Australian university. This very problem was on the first problem set I entered. It’s quite easy to prove with induction (the method I used,) or with reductio ad absurdum.

<spoiler>:
It is always possible.

Yes, the answer and the proof are quite easy (only very basic geometry required) – once you have seen it.

So did you get it right when you did the competition?

does topology count as geometry

I don’t know, but I wouldn’t call the given proof topology (but that may be because I don’t really know how topology is defined).

I will reveal that “yes” is the correct answer.

But the simple proof is the interesting bit.

The Rev Dodgson said:

btm said:

The Rev Dodgson said:

… to this puzzle exists (but I didn’t see it).

Suppose you have an equal number of red and blue dots on a sheet of paper, and no three dots are co-linear.

Is it always possible to connect all the dots in red-blue pairs with straight lines, so that no two lines intersect?

Prove it.

Sydney Uni Maths Society used to (and maybe still do) have an annual maths competition, consisting of ten problems each, available to anyone enrolled at any Australian university. This very problem was on the first problem set I entered. It’s quite easy to prove with induction (the method I used,) or with reductio ad absurdum.

<spoiler>:
It is always possible.

Yes, the answer and the proof are quite easy (only very basic geometry required) – once you have seen it.

So did you get it right when you did the competition?

Yes.

Wouldn’t it depend on how the dots are packed?

wookiemeister said:

Wouldn’t it depend on how the dots are packed?

No, other than that no 3 points can form a straight line.

I will be revealing the simple solution to all after dog walking duties are done, so if anyone wants to have a go at it, you’d better get a move on.

OK, don’t say you weren’t warned.

For any collection of connected points, there will be at least one that has the minimum total length of all the segments. There may be more than one, but there must be just one minimum length.

If the points are connected so that the total length is a minimum, then we can be sure that there are no intersecting lines.

How can we be sure of that?

Because if any two lines do intersect the four points can be reconnected so that they don’t intersect, and it can be shown with simple geometry that the non-intersecting lines have a shorter total length than the intersecting lines.

So if the collection of connected points has any intersecting lines, that collection does not have the shortest total length.

So the shortest total length collection does not have any intersections.

The Rev Dodgson said:

if any two lines do intersect the four points can be reconnected so that they don’t intersect,

stop right there why is there more than that

SCIENCE said:

The Rev Dodgson said:

if any two lines do intersect the four points can be reconnected so that they don’t intersect,

stop right there why is there more than that

It’s possible that when you reconnect those four points, the two new lines will intersect other lines.

The Rev Dodgson said:

SCIENCE said:

The Rev Dodgson said:

if any two lines do intersect the four points can be reconnected so that they don’t intersect,

stop right there why is there more than that

It’s possible that when you reconnect those four points, the two new lines will intersect other lines.

fine well we don’t know if the overall solution there was as obvious as you say then but hey we’ren’t geometers so we’ll take it

SCIENCE said:

The Rev Dodgson said:

SCIENCE said:

stop right there why is there more than that

It’s possible that when you reconnect those four points, the two new lines will intersect other lines.

fine well we don’t know if the overall solution there was as obvious as you say then but hey we’ren’t geometers so we’ll take it

The site where I read about it went into more detail about the proof :)

(and it certainly wasn’t obvious to me before I read the solution).

The Rev Dodgson said:

OK, don’t say you weren’t warned.

For any collection of connected points, there will be at least one that has the minimum total length of all the segments. There may be more than one, but there must be just one minimum length.

If the points are connected so that the total length is a minimum, then we can be sure that there are no intersecting lines.

How can we be sure of that?

Because if any two lines do intersect the four points can be reconnected so that they don’t intersect, and it can be shown with simple geometry that the non-intersecting lines have a shorter total length than the intersecting lines.

So if the collection of connected points has any intersecting lines, that collection does not have the shortest total length.

So the shortest total length collection does not have any intersections.

Oh, that’s clever!

mollwollfumble said:

The Rev Dodgson said:

OK, don’t say you weren’t warned.

For any collection of connected points, there will be at least one that has the minimum total length of all the segments. There may be more than one, but there must be just one minimum length.

If the points are connected so that the total length is a minimum, then we can be sure that there are no intersecting lines.

How can we be sure of that?

Because if any two lines do intersect the four points can be reconnected so that they don’t intersect, and it can be shown with simple geometry that the non-intersecting lines have a shorter total length than the intersecting lines.

So if the collection of connected points has any intersecting lines, that collection does not have the shortest total length.

So the shortest total length collection does not have any intersections.

Oh, that’s clever!

Yes, I thought so.

It came from a mathematical guy on Quora whose posts are usually a mystery to me.

The Rev Dodgson said:

mollwollfumble said:

The Rev Dodgson said:

OK, don’t say you weren’t warned.

For any collection of connected points, there will be at least one that has the minimum total length of all the segments. There may be more than one, but there must be just one minimum length.

If the points are connected so that the total length is a minimum, then we can be sure that there are no intersecting lines.

How can we be sure of that?

Because if any two lines do intersect the four points can be reconnected so that they don’t intersect, and it can be shown with simple geometry that the non-intersecting lines have a shorter total length than the intersecting lines.

So if the collection of connected points has any intersecting lines, that collection does not have the shortest total length.

So the shortest total length collection does not have any intersections.

Oh, that’s clever!

Yes, I thought so.

It came from a mathematical guy on Quora whose posts are usually a mystery to me.

Glad one of them made sense then. Sounds like you finally have him worked out.

roughbarked said:

The Rev Dodgson said:

mollwollfumble said:

Oh, that’s clever!

Yes, I thought so.

It came from a mathematical guy on Quora whose posts are usually a mystery to me.

Glad one of them made sense then. Sounds like you finally have him worked out.

No, he was just keeping things simple for us non-mathemagicians for a change :)

The Rev Dodgson said:

roughbarked said:

The Rev Dodgson said:

Yes, I thought so.

It came from a mathematical guy on Quora whose posts are usually a mystery to me.

Glad one of them made sense then. Sounds like you finally have him worked out.

No, he was just keeping things simple for us non-mathemagicians for a change :)

Long may this last. ;)