# 204 geometry/touching.blocks.p

## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 204 geometry/touching.blocks.p

Can six 1x2x4 blocks be arranged so that each block touches n others, for all n?

geometry/touching.blocks.s

n=0: 6 separate blocks

n=1: 3 pairs

n=2: 2 threesomes

n=3: a 3x3 grid

n=4: a box (each sides touches the four adjoining sides, but not the opposite)

n=5:

Crude ascii:

Front view: Side view:

WWBBBBWWWBBBBWWWBBB

WWBBBBWWWBBBBWWWBBB

:::::::::::::::::::

WWBBBBWWWBBBBWWWBBB

WWBBBBWWWBBBBWWWBBB

To show this happens in general: because the width of the rectangle is a

non-multiple of b, it is possible to position it on the pattern so that the

leftmost column in the rectangle is white and the column just right of the

right edge of the rectangle is black. Suppose N columns are black with this

positioning. Then the rectangle contains N*H black cells, where H is the

height of the rectangle.

If we then shift the rectangle right by one, the number of black columns

increases by 1 and it contains (N+1)*H black cells. The difference between

these two numbers of black cells is H, which is not a multiple of a.

Therefore N*H and (N+1)*H cannot both be multiples of a, and so one of these

two positionings of the pattern will suit your purposes.

David Seal

dseal@armltd.co.uk

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