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## Challenge 260: Solo Scramble

Hazel is playing a solo variant of scrabble.

She plays on an infinite square grid and creates arrangements of tiles on the grid using the following rules:

(i) all tiles must be connected, with at least one side in common with another tile and no set of tiles separate from any other set of tiles

(ii) any line of adjacent tiles, either horizontal or vertical, must spell out a word.

She calls each arrangement she makes a “scramble”.

Hazel doesn’t know very many words and doesn’t want to keep looking things up in the dictionary, so she decides on the following rules:

1. All words have exactly three letters
1. A string of three letters is a word if and only if the middle letter is a vowel (one of a, e, i, o, u) and the first and last letters are consonants.

She takes n vowels and m consonants out of the bag and uses all the tiles to create a scramble.

1. Show that m ≥ n. For what values of m and n can you have m = n?
1. What is the maximum possible value of m / n? For which values of m and n can you achieve this maximum?