Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.PCS 1/10/84; site ahutb.UUCP
Path: utzoo!watmath!clyde!cbosgd!ihnp4!houxm!ahuta!ahutb!leeper
From: leeper@ahutb.UUCP (m.r.leeper)
Newsgroups: net.jokes,net.puzzle
Subject: Re: Re: Re: Short-swords
Message-ID: <535@ahutb.UUCP>
Date: Thu, 7-Mar-85 22:55:13 EST
Article-I.D.: ahutb.535
Posted: Thu Mar  7 22:55:13 1985
Date-Received: Sat, 9-Mar-85 07:13:14 EST
References: <233@tekred.UUCP> <222@wuphys.UUCP> <101@ucbcad.UUCP>, <324@talcott.UUCP>
Organization: AT&T Information Systems Labs, Holmdel NJ
Lines: 57
Xref: watmath net.jokes:11246 net.puzzle:578

REFERENCES:  <233@tekred.UUCP> <222@wuphys.UUCP> <101@ucbcad.UUCP>, <324@talcott.UUCP>

Using finite differences and extrapolating up from lower dimensional
cases, the answer is easy to see, but probably harder to prove.

Dividing up a line with n points you get n+1 pieces.  The first set of
difference are all ones and all after that are zeros.

1
        1
2               0
        1               0
3               0               0
        1               0
4               0
        1
5

Dividing up a plane with lines you get the first set of differences
are the values from the previous case:

1
        1
2               1
        2               0
4               1               0
        3               0               0
7               1               0
        4               0
11              1
        5
16

One can see this is correct because the second cut can intersect the
first cut, making four pieces.  The next cut can intersect two previous
cuts, adding three pieces.  The next cut can interect all three
previous cuts adding four more pieces, etc.

1
        1
2               1
        2               1
4               2               0
        4               1               0
8               3               0               0
        7               1               0
15              4               0
        11              1
26              5
        16
42

The n-th term is (n^3 + 5*n + 6)/6
Twenty cuts will give you 1351 pieces.

                                Mark Leeper
                                ...ihnp4!ahutb!leeper