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Assigned Tuesday September 17; due Friday September 27, 2PM at SS 1071

this document in PDF: HW02.pdf

All of Spivak Chapters 2 and 3.

From Spivak Chapter 2: 1, 5.

From Spivak Chapter 3: 6, 13.

From Spivak Chapter 2: 3, 4, 12, 22.

From Spivak Chapter 3: 1, 7, 21.

We assert that in all sets with precisely horses, all horses are of the same color. For , this is obvious: it is clear that in a set with just one horse, all horses are of the same color. Now assume our assertion is true for all sets with horses, and let us be given a set with horses in it. By the inductive assumption, the first of those are of the same color and also the last of those. Hence they are all of the same color as illustrated below:

From Spivak Chapter 2: 27, 28.

A little more on Chapter 2, Problem 22:

- We know that if and are non-negative then
. This is the same as saying that
,
which is the same as saying that the area of four by rectangles is
less than or equal to the area of a square with side . Can you
actually fit four by rectangles inside a square of side
without overlaps? It's fun and not too hard.
- We know that if , and are non-negative then
. This is the same as saying that
, which is the same as saying that the volume of 27
by by rectangular boxes is less than or equal to the volume
of a cube with side . Can you actually fit 27 such by by
rectangular boxes inside a cube of side without overlaps? This is
also fun, but quite hard. You have no chance of doing it without a physical
model. Make yourself one!
- The corresponding problem in 4D, involving 256 boxes of size
, is actually a little easier, though trickier, than the
3D problem. Can you do it?
- The corresponding problem in 5D, involving 3,125 boxes of size
, is an open problem -- meaning that
nobody knows how to solve it. Can you?

Horse picture from
http://lib.allconet.org/story_hour.htm.

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Dror Bar-Natan 2002-09-30