Home > Polygonal Areas > Description

Polygonal Areas

The Task

Given a positive integer n, create a polygon by selecting n vertices from an n × n grid of points, subject to the following constraints:

  • No two vertices are from the same row or from the same column of the grid.

  • No two sides have the same slope.

  • No two sides intersect, except at a shared vertex.

Here are some examples for n = 6:

Polygon with two vertices in the same row. Polygon with two sides that have the same slope. Polygon with intersecting sides. Valid polygon.
Example 1
Invalid. Two vertices
from the same row.
Example 2
Invalid. Two sides have
the same slope.
Example 3
Invalid. Sides intersect.
Example 4

A Word From Our Sponsor

The Contest

For each integer n from the list below submit two polygons as described above. One of the polygons should have the largest area you can create; the other should have the smallest. See How to Enter, below, for instructions on how to submit your polygons.

The values of n are 5, 7, 11, 17, 23, 29, 37, 47, 59, 71, 83, 97, 113, 131, 149, 167, 191, 223, 257, 293, 331, 373, 419, 467, 521.

For each value of n you can submit more than two polygons if you wish, but only the most extreme – that is, the one with the largest area and the one with the smallest – will count.

See The Scoring System, below, to learn how we determine the winner.

This contest was inspired by Cihan Altay's 2002 Loop Pool puzzle.

The Prizes

First prize: Any item, up to a $500 value, from Bathsheba Sculpture.
Second prize: Any item, up to a $100 value, from the same site.

Al Zimmermann is particularly partial to the metal sculptures, but if you win don't let that influence your choice – the laser crystals are also pretty cool.

How to Enter

Just paste your polygons into the large box on the Submit page and click the Submit Entry button. Format your polygons as follows:

  • An individual polygon consists of a comma-delimited list of points.

  • Each point consists of an x-coordinate and a y-coordinate, separated by a comma and enclosed in parentheses. Coordinates must range from 1 to n.

  • To submit more than one polygon at a time, separate them with semicolons. Do not put a semicolon after your last polygon.

  • Include spaces and line breaks anywhere you like (except within a number) to improve readability.

For example, you could submit the polygon from Example 4 above by entering: (1,2), (2,6), (3,4), (4,5), (6,3), (5,1)

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 polygon you submitted with the largest area and the one with the smallest. The difference between these two areas is your raw score for that n.

  • For each of the 25 possible values of n, we also identify the polygon with the largest area submitted by any entrant and the polygon with the smallest area 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.

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.

Let's walk through a simplified example. Suppose that we modify the contest by asking you to submit polygons for only three values of n: 6, 8 and 10.

Further suppose that we have 3 entrants (Hillary, Donald and Gary) and that these are their most extreme polygons and their raw scores:

n = 6 n = 8 n = 10
Min AreaMax AreaRaw
Min AreaMax AreaRaw
Min AreaMax AreaRaw
Hillary 6.513.5 7.0 14.523.5 9.0 24.534.510.0
Donald 5.511.0 5.5 7.018.511.5 28.040.512.5
Gary 9.0 9.50.5 15.517.01.5

We note the conjectural best for each problem, as follows:

n = 6 n = 8 n = 10
Min AreaMax AreaConjectural
Min AreaMax AreaConjectural
Min AreaMax AreaConjectural
All Entrants 5.513.5 8.0 7.023.516.5 24.540.516.0

Finally, we compute the subscores and contest score for each entrant:

n = 6 n = 8 n = 10 Contest Score
Hillary 7.0 / 8.0 = 0.8750 9.0 / 16.5 = 0.5455 10.0 / 16.0 = 0.6250 2.0455
Donald 5.5 / 8.0 = 0.6875 11.5 / 16.5 = 0.6970 12.5 / 16.0 = 0.7813 2.1657
Gary 0.5 / 8.0 = 0.0625 1.5 / 16.5 = 0.0909 1.5 / 16.0 = 0.0938 0.2472

Donald has the highest contest score and therefore wins.

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 groups.io. 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 mid-contest. In all matters contest-related, my word is final.

Frequently Asked Questions

  • 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 solutions on your own behalf.

    If you would like to form a team, please follow these instructions:

    • If they do not already exist, create individual accounts for each team member. Do not create a second account for any team member who already has one.

    • Let me know each team member's registered email address. It is important that no team member submit any entries to this contest until I notify you that the team has been created and that it is okay to begin submitting.

    • After I've created the team, team members can submit entries to the contest from their individual accounts. The contest engine automatically intercepts these entries and diverts them to the team account. The team is listed on the standings page.

    • Note: Teams are contest-specific. Joining a team for this contest does not affect your participation in other contests.
  • Can I write a program that submits entries by bypassing my browser?

    Yes and no. If your program keeps track of your submissions and only submits improved solutions, 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 individual was single-handedly responsible for 90% of the entries in the database. Please compute responsibly.

  • What topics are appropriate for the discussion group?

    With only one exception, if it's related to AZsPCs then it's fine to talk about it in the discussion group. The exception is spoilers. Spoilers include:

    • specific solutions
    • detailed algorithms

    If you are not sure if something would be considered a spoiler, ask me.

  • After I submit a solution, the scorer shows me the solution's "canonical representation". What is that?

    After the scorer calculates your raw score it may rotate and reflect your polygon, reverse the order of its points, and/or choose a different starting point in order to create a standard (or "canonical") representation. Representing solutions canonically makes it easier to notice when two seemingly different solutions are fundamentally the same.

  • My cousin, who was 1st runner-up in last year's Miss Parador contest, would like to know if you are married. Are you?

    Please ask her to email me privately.