# Tile Edge Challenge

I've been having a problem with these problems for a week now so I thought I'd post it on MSE. Read the text below. . I've also typed it up:

Annabel made a shape by placing identical square tiles in a frame as shown in the diagram above. The tiles are arranged in columns. Each column touches the base but no columns touch the side or top. There are no empty gaps between columns. The frame can be enlarged as needed.

These are the questions. I've also typed them up. A. Show that it is possible to arrange $7$ tiles so that the ant walks $8,9,10,11,12,13,14,15$ tile edges.

B. Show 6 ways of arranging 7 tiles so that the ant walks a total of 9 tile edges.

C. Show it is possible to arrange 49 tiles so that the ant walks a total of less than 21 tile edges.

D. Show four arrangements of 137 tiles, each arrangement with a different maximum height, so that the ant walks a total of 34 tile edges.

For question $a$, is there a formula relating the number of tile edges exposed to the number of tiles. These are my answers for $a$  For the above picture, it's the left hand side which has the $a$ answers.

However, I'm having a bit of trouble with $b$. I've already got four answers down but where are the other two. My four current answers are in the right hand side of the last image. Also, I'll also appreciate a formula to show the relationship.

I'm also having trouble with $c$. I found out that you get 21 tile edges exposes when you make a $7*7$ square but the answer is asking for less than 21. Can anyone help me. Also, as in the previous questions, a formula to show the relationships would be great.

And for $d$, I am totally lost. I have no idea where to start, and I certainly can't find out four arrangements. So can anyone help me? And as always, a formula is nice

P.S. Please explain this in an understandable formula, as I'm only a Year 7 with the ability to understand linear algebra and parts of quadratics and trigonometry. Also, sorry about the poor imaging.

• My I sight must have been worsening....
– zoli
May 9, 2017 at 12:02
• Can you type out your questions please. It shows you have put in some effort, and means everyone can read them and they show up in searches May 9, 2017 at 12:06
• Please add some better quality photos of the exercise itself or just type it in. As of now, it is pretty hard to understand them (using our sight) May 9, 2017 at 12:13
• A base of five tiles, with two tiles on top, should give you 9. May 9, 2017 at 12:42

If the shape is convex, since the height ant go up equal to the height ant go down and side length ant go is side, therefore Formula of convex type is 「Total=HighestVertical×$2$+LongestSide」

b: Lost pieces are $(3,3,1)(1,3,3)$.

c: $(5,5,5,5,5,5,5,5,5,4)$ is $20$.

d: Here are 4 ways $(7,7,,,7,4) (8,8,,,8,1) (9,9,,,9,2) (10,10,,,10,7)$.

• I've already got those answers for b but dont worry I've got an answer. Also, I dont get the formula. "The highest vertical"*2+ The amount of sides of what?" And how did you get this formula
– bio
May 9, 2017 at 22:44
• I really need some commas and correct grammar for the question you asked me. Also, you can have squares at the top of the base. For example, the ant can walk 5 up, 2 across, 1 up, 4 across, 6 down
– bio
May 10, 2017 at 8:46
• Also, c gives me 25 tile edges
– bio
May 10, 2017 at 9:40
• @bio C is 5 up, 9 across, 1 down, 1 across, 4 down, then $5+9+1+1+4=20$. May 10, 2017 at 10:06
• Oh sorry I though you were going 10 up, 4 across, 1 down, 1 across, 9 down
– bio
May 10, 2017 at 10:31