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### Chess Problem

```
Date: Wed, 16 Nov 94 13:21:31 GMT
From: "Nathaniel Chell"
Subject:  Urgent Problem

Hi,

Dr. Math, we are investigating the number of squares on a chess board
"controlled" by the Queen at any one position, i.e., where she could move
(anywhere in a straight line).

On a standard chess board we found a pattern using "concentric squares"
or "rings"

----------------------------------------
| 1    2    3    4    5    6    7    8 |
|    ------------------------------    |
|    |                            |  <-+-- controls 22 sqaures
|    |                            |    |
|    |   ---------------------- <-+----+-- controls 24 squares
|    |   |                    |   |    |
|    |   |                  <-+---+----+-- controls 26 squares
... and so on ...

We have the equation 3x-2, where x = # of squares along top row.
So if the Queen is on the top row of an 8x8 board she controls 22
squares.

When the queen moves down to the second row, we use the same formula,
but attach "+2" on the end, ie. (3x-2)+2 / (3x-2)+2+2 / (3x-2)+2+2+2,
and so on as the Queen moves down the rows.

So... We were wondering if you guys could help us with an equation which
works wherever the queen is, without having to keep on adding twos,
which is messy and could get confusing.

Many, many, many thanks in advance,

Nathaniel Chell
Wesley Ball
David Gould

Sir Frederic Osborn School, Welwyn Garden City, UK.
```

```
Date: Wed, 16 Nov 1994 10:57:51 -0500 (EST)
From: Dr. Sydney
Subject: Re: Urgent Problem

Dear Nathaniel, Wesley, and David,

Thanks for writing Dr. Math with your urgent question.  You seem to
have a very good handle on the problem you are working with.  Indeed, the
number of squares the queen controls does form a pattern of "concentric
squares."  I'm not quite sure what you are asking us for--did you want a
formula in which you could plug in the coordinates of the queen to get the
number of squares she controls?  It seems to me like you have a pretty nice
formula already--just determine which "concentric square" the queen is in,
and then determine the number of squares she controls based on that.  If you
are looking for something more, please write back and we will try to help.
Feel free to write back with other questions, too -- we love to hear from
you!

--Sydney, "Dr. Chess"
```

```
Date: Wed, 16 Nov 94 18:08:26 GMT
From: "Nathaniel Chell"
Subject:  Re: Urgent Problem

Cheers Sydney,

We will continue with the method of determining which concentric
square she's in and then use the previous formula. However, if you
could help us with the more complex way of doing it we may be able
to get into the higher GCSE bands/grades. Is it possible?? Your help is
much, much appreciated. And now we know about you guys we'll have
a few more problems for you in the future as we come closer to our GCSEs.

Thanks again,

Nat, Dave & Wes.
__________

Date:     Thu, 17 Nov 94 18:32:06 GMT
From: "Nathaniel Chell"
Subject:  Attn: Dr. Chess

Hello again,

As you may remember we asked again about a more complex formula
than 3x-2 in our chess board problem. We've just gone one further:
3n+2(r-1), where n = number of squares along top row, and
r = ring / concentric square the Queen is in.

Wesley figured it out. We thank you for your help, and if you boffs
can come up with any better, *please* let us know.

Otherwise, thanks again,

Nat, Dave & Wes.
Form 11D.
Sir Frederic Osborn School, Welwyn Garden City, UK.
```
Associated Topics:
Middle School Puzzles

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