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Initially Billy is given two options "A-M" or "N-Z". By hitting the "right arrow", two nodes appear under "N-Z" further dividing the letters into "N-T" or "U-Z". This could continue until Billy ends up at only one letter, which would require 4-5 key presses. The design shown above narrows down to three letter, at which point Billy could use the "left", "right", "select" keys to choose from one of three possible letters. After a letter is selected, the visualization returns to the root nodetree clears leaving only the first two choices.
Binary Search Letter Selection List
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If Billy is required to halve until one letter remains, this would take 4-5 selections. It might be more efficient to make two cuts so three letters remain, and let Billy choose from one of the three. People who play video games often remember short key sequences as combination moves in games, and might be quick to remember the key presses for word combinations.