A Bidirectional Search

Date: 01/27/2003 at 23:28:06
From: Nait
Subject: How can I get the numbers from 1-100 using only 4 eights?

1 = 8 divided by 8 = 1 plus 8 minus 8, and so on, for all the numbers
from 1 to 100. I have got many of them and I need to get the others. I
just can't figure them out.

Thanks so much for your help.

Date: 01/29/2003 at 17:59:49
From: Doctor Edwin
Subject: Re: How can I get the numbers from 1-100 using only 4 eights?

Hey, Nait!

In a few pages in a notebook, write the numbers from 1 to 100 going 
down the left side. Now, instead of trying to get a specific number, 
just start playing with the 8's. 8+8+8+8? 32. Write 8+8+8+8 down next 
to 32. Put a check mark down next to 32. One down, 99 to go. 8+8+8-8? 
Write it down next to 16 and check that one off. 88/8-8? (88-8)/8? 
Just keep writing them down next to their values. Try to be 
systematic, like trying everything you can think of with one 88 before 
moving on to some other approach.

An important thing to do is to record all your intermediate results. 
This will help you a lot. When you did 88/8-8, you had 11 there for a 
second, right? So next to 11, write down 88/8, but don't check off 11 
(since you haven't solved 11 yet). Maybe you want to circle the ones 
that use four 8's, or maybe keep a separate list for intermediate 
results. I know it sounds like a lot of bookkeeping, but here's how 
it helps:

You've played around and you've got maybe 60 numbers checked off and 
things are slowing down. You've tried most of the things you can 
think of, but you keep finding other ways to get the same numbers. So 
now you start thinking about the numbers you haven't gotten yet. Like 
19, for example. So you look at 19, and you wonder if it's 8 away 
from something you know how to make with three 8's. "Hey, I can make 
11 with 3 8's, and so I add 8 and now I write down 88/8+8 next to 19 
and check it off."

We call this "bidirectional search." You're not just going from ways 
of combining 8's to numbers you need, and you're not just going from 
numbers you need to ways of combining 8's. You're working from both 
ends at the same time and that shortens your search by a LOT.

When you show this project to your teacher, she'll be impressed.

- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/