# To all you math dopers

**URL:** <https://boards.straightdope.com/t/to-all-you-math-dopers/124958>\
**Category:** Factual Questions\
**Created:** [August 21, 2002, 5:43pm UTC](https://boards.straightdope.com/t/to-all-you-math-dopers/124958 "2002-08-21T17:43:06Z")\
**Posts on this page:** 8\
**Page:** 1

<div class="post-metadata">

**Author:** ![High\_Deity](https://avatars.discourse-cdn.com/v4/letter/h/e9a140/32.png) [@High\_Deity](https://boards.straightdope.com/u/High_Deity)\
**Post date:** [August 21, 2002, 5:43pm UTC](https://boards.straightdope.com/t/to-all-you-math-dopers/124958/1 "2002-08-21T17:43:06Z")

</div>

Here’s one for you:

You have 5 squares to put in a pattern with the following 2 rules:

1. at least 1 complete side of a square must be in contact with 1 complete side of another square.
2. no rotations or mirror images allowed.

how many patterns can be formed?

no trick questions here. no hidden answers. actually, i’m gonna give you the answer, because answering the top problem isn’t the big concern. the answer is 12.

what i want to know, is given number of squares _n_, what is the formula that tells you the number of distinct patterns that can be formed?

---

<div class="post-metadata">

**Author:** ![emarkp](https://avatars.discourse-cdn.com/v4/letter/e/3be4f8/32.png) [@emarkp](https://boards.straightdope.com/u/emarkp)\
**Post date:** [August 21, 2002, 6:02pm UTC](https://boards.straightdope.com/t/to-all-you-math-dopers/124958/2 "2002-08-21T18:02:13Z")

</div>

I assume the squares are all the same size, right? And that rotating individual squares doesn’t change the pattern?

---

<div class="post-metadata">

**Author:** ![High\_Deity](https://avatars.discourse-cdn.com/v4/letter/h/e9a140/32.png) [@High\_Deity](https://boards.straightdope.com/u/High_Deity)\
**Post date:** [August 21, 2002, 6:03pm UTC](https://boards.straightdope.com/t/to-all-you-math-dopers/124958/3 "2002-08-21T18:03:59Z")

</div>

yes, all squares are the same size. we can just say 1 unit by 1 unit to make things simple.

---

<div class="post-metadata">

**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\
**Post date:** [August 21, 2002, 6:10pm UTC](https://boards.straightdope.com/t/to-all-you-math-dopers/124958/4 "2002-08-21T18:10:37Z")

</div>

I found 14 arrangements that fullfil your requirements. There are two arrangements that you didn’t count. (And I know which two. 🙂 )

I found them by brute force; I don’t know the solution to the general problem.

---

<div class="post-metadata">

**Author:** ![NotMrKnowItAll](https://avatars.discourse-cdn.com/v4/letter/n/6bbea6/32.png) [@NotMrKnowItAll](https://boards.straightdope.com/u/NotMrKnowItAll)\
**Post date:** [August 21, 2002, 6:29pm UTC](https://boards.straightdope.com/t/to-all-you-math-dopers/124958/5 "2002-08-21T18:29:48Z")

</div>

I got the fourteen, but I question if **a** pattern can be disjoint. I may have a recursive solution, maybe not.

---

<div class="post-metadata">

**Author:** ![Omphaloskeptic](https://avatars.discourse-cdn.com/v4/letter/o/bcef8e/32.png) [@Omphaloskeptic](https://boards.straightdope.com/u/Omphaloskeptic)\
**Post date:** [August 21, 2002, 7:10pm UTC](https://boards.straightdope.com/t/to-all-you-math-dopers/124958/6 "2002-08-21T19:10:34Z")

</div>

> [@](#):
>
> \*Originally posted by High Deity \*  
> \*\*Here’s one for you:
> 
> You have 5 squares to put in a pattern with the following 2 rules:
> 
> 1. at least 1 complete side of a square must be in contact with 1 complete side of another square.
> 2. no rotations or mirror images allowed.
> 
> how many patterns can be formed?
> 
> no trick questions here. no hidden answers. actually, i’m gonna give you the answer, because answering the top problem isn’t the big concern. the answer is 12.
> 
> what i want to know, is given number of squares _n_, what is the formula that tells you the number of distinct patterns that can be formed? \*\*

These shapes are called **polyominos** (see [here](http://www.mathpages.com/home/kmath039.htm)). (With 5 squares, they’re called pentominos, etc.) There’s no simple expression known for the number of polyominos.

---

<div class="post-metadata">

**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\
**Post date:** [August 21, 2002, 8:21pm UTC](https://boards.straightdope.com/t/to-all-you-math-dopers/124958/7 "2002-08-21T20:21:27Z")

</div>

I have a set of pentominoes - you can make quite a few interesting shapes wtih them, both lying flat and 3D shapes.

---

<div class="post-metadata">

**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [August 21, 2002, 10:48pm UTC](https://boards.straightdope.com/t/to-all-you-math-dopers/124958/8 "2002-08-21T22:48:01Z")

</div>

Some bounds are given [here](http://mathworld.wolfram.com/Polyomino.html).