[bonanova]

The story goes that a man bequeathed a square plot of land to his five sons.

His first son was his favorite, so he got 1/4 of the land - a square part in the North-West corner.

That left an L-shaped piece to the East and South. Wanting the other four sons to have

equivalent parcels, equal not only in area but also in shape, he's asking you to come up

with a way to carve that L-shaped piece into 4 congruent parcels.

See if you can come up with the solution.

OK, so that's not the puzzle I have for you.

Let's suppose there were no favored son, and the father wanted all five

of his sons to have parcels of exactly the same area and exactly the same shape.

Can you solve that one? Leave the spoiler open if that will help.

[itachi-san]

This seems too easy to be right, but:

Just draw 4 evenly spaced lines horizontally or vertically across the total area. This will give 5 plots of the same shape and same area.

[Guest]

I take it the buildup was just a red herring?

There is, naturally, a more complicated variant. The father wants every son to share a border with every other son. Is it possible?

Edited April 4, 2008 by Duh Puck

[bonanova]

The red herrings are all neatly caught in the net.

Good job.

[bonanova]

There is, naturally, a more complicated variant. The father wants every son to share a border with every other son. Is it possible?

I take it the buildup was just a red herring?

I'm thinking it's not possible - without at least one bridge.

It's like connecting each of 5 dots directly to the other 4 without a crossing.

[Guest]

There is, naturally, a more complicated variant. The father wants every son to share a border with every other son. Is it possible?

I take it the buildup was just a red herring?

Question: If 4 squares are together, lets say:

red, blue

green, yellow

Is yellow and red consider to be sharing a border, as in diagonals...

Edited April 4, 2008 by PolishNorbi

[Guest]

Question: If 4 squares are together, lets say:

red, blue

green, yellow

Is yellow and red consider to be sharing a border, as in diagonals...

No. They have to share an edge, not just a point.

[Guest]

I'm thinking it's not possible - without at least one bridge.

It's like connecting each of 5 dots directly to the other 4 without a crossing.

Just to wrap this one up since I left it open ...

Correct. They cannot all share an edge. I remember reading that problem when I was just a kid (10 years old) in a book called "Bet You Can't!", and when I saw your problem I at first assumed it was the same question and posted an incorrect answer before catching my mistake and pulling a complete about-face with an edit.

As I recall, the book explained that this principle (of five regions not being able to share a common edge) was why maps could make do with only five colors without ever having two regions of the same color border each other. I haven't verified that, but it seems reasonable.

[Guest]

Just to wrap this one up since I left it open ...

Correct. They cannot all share an edge. I remember reading that problem when I was just a kid (10 years old) in a book called "Bet You Can't!", and when I saw your problem I at first assumed it was the same question and posted an incorrect answer before catching my mistake and pulling a complete about-face with an edit.

As I recall, the book explained that this principle (of five regions not being able to share a common edge) was why maps could make do with only five colors without ever having two regions of the same color border each other. I haven't verified that, but it seems reasonable.

Actually, this is a famous problem of mathematics called the four-color map theorem. I am not sure how old it is, but I believe a couple of hundred years, or so. The theorem states that it is possible to color any map, no matter how complex, with only four colors, such that no two adjacent regions share the same color. The problem was nobody could either prove it, or come up with a counter-example, It was solved in the 1970's by proving that the theorem was true. The proof was done by computer and took several thousand pages to print out. It was a bit of a scandal for a while , because no one could check the proof for accuracy and many mathematicians refused to accept the proof at all, saying that such a proof was non-standard and un-checkable, even in principle. But others hailed it as the first of a new kind of systematic approach to doing math. And, of course, that view has prevailed.