# Luminescence Bugs

Thirty rare species of insect were trapped in a net and placed in two jars marked A & B.

All types of these lamp beetles (Red, Green and Blue) inside a jar are always glowing with the same color according to the most number of beetle type within the jar.

In case the most has equal in numbers they all glow with the same color mixture.

By scooping some of the beetles from one jar and transferring them to the other jar (they may escape in the process), we can know the number of each beetle type from the results of the 3 trials shown below.

How many Red, Green and Blue beetles remain in both jars?

Let $(R,G,B)$ be the number of red, green, and blue beetles in a jar. I deduced the following jar contents:

$(5,2,0)$, $(7,8,8)$
$(8,4,8)$, $(4,6,0)$
$(2,0,4)$, $(10,10,4)$
$(4,4,4)$, $(7,6,4)$,

which I think checks out. The final counts can be obtained without using much of the information given.

Label the jars by rows 1-4 and columns A-B. Let $(R,G,B)$ be the total beetle counts and $(R_i,G_i,B_i)$ be the counts for Jar $i$. Jar 4A is white and has 12 beetles and therefore must have $(4,4,4)$. Jar 4B has 17 beetles; if $R_{4B}\le6$, then it could have at most $6+5+5=16$ beetles, so $R_{4B}\ge7$. Then $R=4+R_{4B}+1\ge12$.

If $R_{3A}\ge3$, then $B_{3A}\le6-R_{3A}\le3\le R_{3A}$ and 3A would not be blue, so $R_{3A}\le2$. Then $R_{3B}=R-R_{3A}\ge10$, so because 3B is yellow, $G_{3B}=R_{3B}\ge10$. Then $G\ge G_{3B}\ge10$.

Because Jar 1B is cyan and has $23$ beetles, $B\ge B_{1B}\ge\frac{23}3>7\implies B\ge8$.

Then $R+G+B\ge30$, so we must have equality, i.e., $(R,G,B)=(12,10,8)$. This already gives us the final jar counts $(4,4,4)$ and $(7,6,4)$.

The other counts can be obtained through further deduction.

• Sorry, I did not get it fully, but was there a case of a escaped beetle? As the final count in both the jars sums up to 29, instead of 30. – Mea Culpa Nay May 13 '18 at 5:54
• @MeaCulpaNay Yes, you can see one red beetle outside Jar 4B to its upper right. – noedne May 13 '18 at 8:31
• Thanks. I got it. It required additional observational skills as well in addition to problem solving skills. – Mea Culpa Nay May 13 '18 at 9:29