# Minimum number of colors required to color letters Pair

Mala has a coloring book in which each English letter is drawn two times.She wants to paint each of these 52 prints with one of the $k$ colors,such that the color pairs used to color any two letters are different.Both prints of the same color can be colored with the same color.What is the minimum number of $k$ that satisfies this requirement?

My progress so far : there is a pair of each letter which can be colored with same color. i.e. $k \cdot k =k^2$ ways to color pair of letters . But Now I don't find way to tackle it further. What is the connection b/w total letters pairs and $k^2$??

More Progress: I came to a conclusion ,see if we have k colors and 26 letters ,and we need every pair of colors to be different ,$k >=26$. But now we have $k^2$ instead of K ,hence $k^2>=26$ might be solution!!

More Progress:But it seems problem of coloring AA,BB.....ZZ is no different than A,B...Z. answer might be $k>=26$

• Is {Red, Blue} and {Blue, Red} considered the same or different? – Masked Man Feb 11 '14 at 13:34
• @Happy I indeed don't know ,Qus is very ambiguous in it's detail. But what if they are considered same and what if not? How would you solve it? Although what Ross Millikan has considered is that they are same. – Rishi Feb 11 '14 at 14:25

Given $k$ colors, we can color $k$ letters with both occurrences having the same color. We can also color ${k \choose 2}=\frac 12k(k-1)$ letters with two different colors. We need $k+\frac 12k(k-1)=\frac 12k(k+1) \ge 26$, so $k=7$ is sufficient.
• ${k \choose 2}=\frac 12k(k-1)$ This might including first steps result also,in which you have calculated n different colors for n letters ,don't you thing we need to minus n from final result? – Rishi Feb 9 '14 at 8:54
• No, the $k-1$ factor says we made the second one different from the first. The factor $\frac 12$ is because red/blue is the same as blue/red. It is the standard result of the number of ways to selec two different colors from $k$ – Ross Millikan Feb 9 '14 at 15:42