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The quick brown …

We arrange the 26 letters of the alphabet in a row as follows:

THEQUICKBROWNFXJMPDVLAZYGS (*)

Now take any letter, say P, and find the longest chains in (*) in alphabetical order ending with P. We find some of length 5, for example EIKMP, but none any longer. Next we find the longest chains in (*) in reverse alphabetical order starting with P. We find some of length 3, for example PLG, but none any longer. We say P has alphabetical length, α=5, and reverse alphabetical length, ω=3.

Question 1: I have chosen a letter which comes before P in the alphabet and to the left of P in (*). Can you say for certain my letter has α less than 5?

Question 2: I have arranged the 26 letters in a new row (**). Can you say for certain that if you choose a letter in (**) and I choose a different letter in (**) then they will have different α’s or different ω’s?

Now I want you to write on a piece of paper, a list of 25 possibilities for α and ω so [α=1 ω=1], [α=1 ω=2], …, [α=1 ω=5], [α=2 ω=1], …, [α=2 ω=5], [α=3 ω=1], …, [α=5 ω=5].

Next I want you to take each letter in (*) and work out α and ω for it and mark it on the list, for example you will write P against [α=5 ω=3]. Unfortunately, some letters wil have a combination of α and ω that is not on the list, for instance X has α=6 and ω=4.

Question 3: Can you arrange the 26 letters of the alphabet in a row so that every letter has a combination of α and ω that is in the list? If your answer is “yes” then give such a row.

Question 4: Can you be certain that you will be able to find in my row (**) a chain of 6 letters that are in either alphabetical or reverse alphabetical order?

A £10 book token will be awarded to the sender of the first correct answer opened on Thursday 28 September. The Editor’s decision is final. Please send entries to Enigma 837, Â鶹´«Ã½, King’s Reach Tower, Stamford Street, London SE99 0BB. The winner of Enigma 831 was Andrew Plant of Lymington, Hampshire.

Answer to Enigma 831

Pot Black

(1) 14 and 13; (2) Whyte.