What are Delacorte Numbers?
Take an m × n rectangle and divide it into an
array of unit cells. Then number the cells, in any order you wish,
from 1 to mn. For example, if your rectangle is 2 ×
3, it might look like this after you've numbered it:
The Delacorte number is an integervalued property
of such a rectangle. It is computed as follows.
For each distinct pair of integers (a, b)
in the rectangle, calculate
D_{a,b} using the formula
D_{a,b}
=
gcd(a, b)
×
distance^{2}(a, b)
where gcd(a, b) is the greatest
common divisor of integers a and b, and
distance(a, b) is the physical
distance (within the rectangle) from integer a
to integer b.
The Delacorte number for the rectangle is the sum of all the
D_{a,b}.
For example, for the rectangle above the calculation of
its Delacorte number looks like this:
a 
b 
gcd(a, b) 
distance^{2}(a, b) 
D_{a,b} = gcd(a, b) × distance^{2}(a, b) 
1  2  1  2  2 
1  3  1  1  1 
1  4  1  1  1 
1  5  1  1  1 
1  6  1  2  2 
2  3  1  1  1 
2  4  2  5  10 
2  5  1  1  1 
2  6  2  4  8 
3  4  1  4  4 
3  5  1  2  2 
3  6  3  5  15 
4  5  1  2  2 
4  6  2  1  2 
5  6  1  1  1 

Delacorte number  53 
The Contest
For each integer n from 3 to 27 submit two
n × n squares numbered with the integers
from 1 to n^{2}. One of the squares should have
the largest Delacorte number you can make; the other should
have the smallest. See How to Enter,
below, for instructions on how to submit your squares.
For each value of n you can submit more than two
squares if you wish, but we will count only your two most
extreme squares – that is, only the square with the
largest Delacorte number and the square with the smallest.
See The Scoring System, below, to
learn how we determine the winner.
My thanks to Marcos Donnantuoni, whose "multiples graph" inspired this contest.
The Prizes
First prize is a $500 gift certificate redeemable at
Bathsheba Sculpture.
Second prize is a $100 gift certificate.
How to Enter
Just paste your squares into the large box on the
Submit page and click the Submit
Entry button. Format your squares as follows:

An individual square consists of a commadelimited list of rows.

Each row consists of a commadelimited list of integers,
enclosed in parentheses.

If you would like to submit more than one square at a time
(either for the same value of n or for different
values) separate them with semicolons. Do not put a
semicolon after your last square.

Include spaces and line breaks anywhere you like (except within a
number) to improve readability.
For example, if you were permitted to submit the nonsquare example above,
you might enter:
(4,1,3), (6,5,2)
Do not submit squares under more than one account. This is important.
Do not submit squares under more than one account.
The Scoring System
The entrant with the highest contest score wins.
Here’s how we calculate your contest score:

For each of the 25 possible values of n, we
identify the square you submitted with the
largest Delacorte number and the one with
the smallest. The difference between these
two Delacorte numbers is your raw score for
that n.

For each of the 25 possible values of n, we
also identify the square with the largest Delacorte
number submitted by any entrant and the square with
the smallest Delacorte number submitted by any entrant.
The difference between those two numbers is
the conjectural best for that n.

Then, for each n, we calculate your subscore
for that n by dividing your raw score by the
conjectural best.

Finally, we calculate your contest score by
adding up your 25 subscores.
Let's walk through a simplified example. Suppose that we modify
the contest by asking you to submit squares of only 3 sizes:
3 × 3, 4 × 4 and 5 × 5.
Further suppose that we have 3 entrants (Heisenberg,
Pinkman and Crystal) and that these are their most
extreme Delacorte numbers and their raw scores:

3 × 3 
4 × 4 
5 × 5 
Min  Max  Raw Score 
Min  Max  Raw Score 
Min  Max  Raw Score 
Heisenberg 
134  187  53 
1121  2071  950 
3933  4501  568 
Pinkman 
145  208  63 
975  1716  741 
4049  5295  1246 
Crystal 
120  175  55 
1083  1908  825 
4135  6006  1871 
We note the conjectural best for each problem, as follows:

3 × 3 
4 × 4 
5 × 5 
Min  Max  Conjectural Best 
Min  Max  Conjectural Best 
Min  Max  Conjectural Best 
All Entrants 
120  208  88 
975  2071  1096 
3933  6006  2073 
Finally, we compute the subscores and contest score for each entrant:

3 × 3 
4 × 4 
5 × 5 
Contest Score 
Heisenberg 
53 / 88 = 0.6023 
950 / 1096 = 0.8668 
568 / 2073 = 0.2740 
1.7431 
Pinkman 
63 / 88 = 0.7159 
741 / 1096 = 0.6761 
1246 / 2073 = 0.6011 
1.9931 
Crystal 
55 / 88 = 0.6250 
825 / 1096 = 0.7527 
1871 / 2073 = 0.9026 
2.2803 
If two entrants have the same contest score, we break
the tie by giving preference to the entrant whose last
improvement was submitted least recently.
Getting Your Questions Answered
First, check the FAQ section below. If you can't find the
information you need there, send your question to the
discussion group. If your question is of a personal nature, and not of
general interest, send an email directly to Al Zimmermann.
The Discussion Group
If you think you might enter the contest, you should join the
contest discussion group. You can join either by sending a blank email
here or by visiting the group on
Yahoo!. The discussion group serves two purposes. First,
it allows contestants to ask for clarifications to the rules. Be aware that
sometimes these requests result in changes to the rules, and the first place
those changes are announced is in the discussion group. Second, the discussion
group allows contestants to interact with each other regarding programming
techniques, results and anything else relevant to the contest.
My Lawyer Would Want Me To Say This
I reserve the right to discontinue the contest at any time. I reserve the right
to disqualify any entry or entrant for any reason that suits me. I reserve the
right to interpret the rules as I see fit. I reserve the right to change the
contest rules in midcontest. In all matters contestrelated, my word is final.
Frequently Asked Questions

Can I enter the contest more than once, using different accounts?
No. Submitting squares from more than one account is not permitted.
In fact, just having more than one account is against the rules.

Can teams enter the contest?
Yes. But a team can only be formed by those who have not
already entered the contest as individuals. Once you enter
as an individual, team membership is no longer open to you.
Likewise, once you've joined a team you can't break away and
start submitting squares on your own behalf. If you would
like to form a team, please email me.

Can I write a program that bypasses my browser and submits
squares directly to AZsPCs over the Internet?
Yes and no. If your program keeps track of your submissions
and only submits improved squares, yes. Otherwise, no. For
the Son of Darts contest a few years ago, there was one participant
who wrote a program to exhaustively generate all possible solutions
and submit every one of them. Within two weeks he'd submitted over
a million entries. The AZsPCs database had grown to 10 times its
normal size and this fellow was singlehandedly responsible for 90%
of the entries in the database. Please compute responsibly.

What information about my squares can I share in the
discussion group?
There are two types of information that you are forbidden to post.
The first is specific squares. The second is code. You may
post scores, so if you want to tell everyone that you got a
raw score of 1,598,259 for n = 20 (whether true or not), go
right ahead. You may also discuss the algorithms you are using.

How can I find out what my individual subscores are?
You can't. I know this is frustrating, but it's a long standing
policy that isn't going to change. Over the years it's been hotly
debated in the discussion group and the contest administrator
appears to have very strong feelings on the matter. You're going
to have to learn to live with it. And I'd think twice before
raising the issue yet again.

After I submit a square, the scorer shows me the
square's "canonical representation". What is that?
After the scorer calculates your Delacorte number it rotates and
reflects your square in order to create a standard
(or "canonical") representation of the square. Having canonical
representations makes it easier to notice when two seemingly
different squares are fundamentally the same. This is particularly
useful at the end of the contest when the best squares become
public.
