# Simple puzzle, but that's not the point - what's the significance of the spacing of the shelves?

We had a cabinet that used to contain CDs. We liked the cabinet but had to admit that we hadn't taken them out for years. We decided to put some shelves in the middle (the outer ones will have hanging pots). My original thought was just to have them equally spaced, but I was told that they needed to be irregular. The shelves are going to contain some of my puzzle collection (see some candidates in the photo), so I thought a somewhat interesting spacing would be worthwhile.

It's probably hard to be certain based only on the picture, but

can you guess the significance of the spacing? If you want to confirm or get a hint, the measurements are below.

Here are the (nominal) heights of the shelves:

From top to bottom. Left side: 6.5, 4.5, 7.5, 4.5, 7.75, 9.25, 5.5, 8, 7.75
From top to bottom. Right side: 5.5, 6.5, 9.25, 8.25, 9.25, 8.5, 7.75, 6.5
These are in inches. I know, I know. But since moving to the US, all my tools are in inches, and I've grown to love being bi-unital. It's like Europeans with languages but easier to translate.

Hint:

If you need a computer (other than to look something up), you're over-thinking. We're mapping length to digits. I was constrained by the height of the cabinet and by aesthetics. But look how many different heights there are.

• Nothing to do with the puzzle (so far as I can see) but was the cabinet originally horizontal? Or were there inserts of some sort to hold the CDs in place? Commented Aug 20, 2021 at 22:54
• No, it was always vertical. There were plastic inserts on both walls design to hold the CDs, and pop them out a bit when you lifted them up. So basically a slot, with a lip at the front and a sprint at the back. Just molded plastic with a mirror image on the other side. It was quite nice, because the space between was empty. I thought of keeping them, maybe even using CD cases for shelves, but without the CDs, they were a bit dark and weird, so I put in the green-sided acrylic shelves (meant to looks like glass and does a pretty good job). Commented Aug 21, 2021 at 6:05
• The integer version of the sequence (multiplied by 4) does not seem to be in the OEIS. Are you sure this is a "simple" puzzle? Commented Aug 23, 2021 at 13:02
• I tried converting the integers (multiplied by 4 as you say) to ASCII character codes and got nowhere either. I think we could do with a hint here! Commented Aug 23, 2021 at 14:04
• Just assigning 1 to 9 to each value in order gives something mighty close to Pi on the left Commented Aug 23, 2021 at 17:51

The shelf heights represent:

the first 17 digits of π: 3.1415926535897932.
@theonetruepath mentioned in the comments above:

Just assigning 1 to 9 to each value in order gives something mighty close to Pi on the left

And in fact, we can extend it to the right-side shelves too.

Find the whole solution below.

Height references from OP:

From top to bottom. Left side: 6.5, 4.5, 7.5, 4.5, 7.75, 9.25, 5.5, 8, 7.75
From top to bottom. Right side: 5.5, 6.5, 9.25, 8.25, 9.25, 8.5, 7.75, 6.5

First:

We map each unique height to a digit from 1-9, with the smallest height being mapped to 1, and so on according to their ascending numerical order:

Height | Mapped to digit
4.5    | 1
5.5    | 2
6.5    | 3
7.5    | 4
7.75   | 5
8      | 6
8.25   | 7
8.5    | 8
9.25   | 9

Then:

We apply the above mapping to the height references:

From top to bottom. Left side: 3, 1, 4, 1, 5, 9, 2, 6, 5 From top to bottom. Right side: 2, 3, 9, 7, 9, 8, 5, 3

The first 17 digits of π are: 3.1415926535897932.

The left-side shelves are simply the first 9 digits of π, left-to-right (in the image top-to-bottom): 3, 1, 4, 1, 5, 9, 2, 6, 5 → 3.14159265.

The right-side shelves are the next 8 digits of π, but you have to read them right-to-left (in the image bottom-to-top), which gives us: 3, 5, 8, 9, 7, 9, 3, 2 → 35897932.

So you start from the top-left shelf, and you go all the way down, then switch to the right-side and go back all the way up to get the first 17 digits of π!

Cool trick! :D