bonanova Posted April 4, 2008 Report Share Posted April 4, 2008 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. Quote Link to comment Share on other sites More sharing options...
0 itachi-san Posted April 4, 2008 Report Share Posted April 4, 2008 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted April 4, 2008 Report Share Posted April 4, 2008 (edited) 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 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted April 4, 2008 Author Report Share Posted April 4, 2008 The red herrings are all neatly caught in the net. Good job. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted April 4, 2008 Author Report Share Posted April 4, 2008 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted April 4, 2008 Report Share Posted April 4, 2008 (edited) 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 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted April 4, 2008 Report Share Posted April 4, 2008 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted April 7, 2008 Report Share Posted April 7, 2008 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted March 8, 2009 Report Share Posted March 8, 2009 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. Quote Link to comment Share on other sites More sharing options...
Question
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.
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.
Link to comment
Share on other sites
8 answers to this question
Recommended Posts
Join the conversation
You can post now and register later. If you have an account, sign in now to post with your account.