# Color Logic - Balance

Inspired by this puzzle

Find the rule that each color chain on the left follows, but none on the right follow. Raw text if you're colorblind or something (O = orange, Y = yellow, G = green, C = cyan):

Left:

COGGYCYOC
GYGYG
O
YYGGGGYYY
OC
GYO
GYYYGOG
OCGCOYO
CYOYOOY
GCGCGCGC


Right:

CC
GCYO
YYYYYOOOO
YCGOYCG
OYCGCY
GOC
CCCGYOO
OOY
CYOGYYCG
GOCOYOGGCO


If you want to see more cases, you may suggest up to 5 additional color chains, and I will sort them accordingly. As a rule of thumb, try to keep the length at $10$ blocks or less. You may suggest chains in this chat.

Hint 1:

If you reverse a chain, it will stay on the same side.

Hint 2:

If you recolor all green to be orange, everything will stay on the same side.

Hint 3:

The rule can be described as the equality of two quantities.

• Another Hint Please? – Holyprogrammer Aug 28 '18 at 6:12
• @KhushrajRathod I will add a hint every 24 hours. – Riley Aug 28 '18 at 11:52
• @Riley Maybe you can add a few 'caterpillar logic' style hints, i.e. we ask a combination and you put it on the left or on the right. – rhsquared Aug 28 '18 at 12:47
• @rhsquared That could work. I'll do that in addition to hints I'm already giving. Not sure what the best way to do this is. Maybe each user is limited to 5 chains or something? – Riley Aug 28 '18 at 12:59
• @rhsquared I've added rules for suggesting more chains to the question. – Riley Aug 28 '18 at 16:26

## 2 Answers

The set on the left is defined by

having equal numbers of adjacent pairs of primary and secondary colored blocks. First, treating O/G and Y/C as equivalent gives a picture like this: Counting a group of n as (n - 1) pairs, all the left sequences have the same number of primary and secondary pairs, while the right sequences differ.

Another way to phrase the answer:

Count the number of primary and secondary colored blocks, weighting each end block .5 less than normal. A solitary block's weighted zero, then. The totals should be the same.

• Yes, you got it! (And right after I added the bounty, too). I will accept this answer and award you the bounty. But not immediately so the question can receive more attention. – Riley Aug 30 '18 at 17:04

This is just putting Mr. Fish's answer into mathematics:

A sequence is valid if, and only if, it consists of two partitions of $n$ (the total number of blocks) into sub-sequences $a$ and $b$, with $\mid a\mid+\mid b\mid=n$. Furthermore, $\mid\operatorname{np}(a)-\operatorname{np}(b)\mid\le1$, where $\operatorname{np}(k)$ gives the number of parts of $k$. Also, the total sums of each part minus $1$ in each sequence are equal.