Task — Missing letter

You are given a sequence of 5 lowercase letters, with one letter replaced by ‘?’.

Each letter maps to its position in the alphabet (‘a = 1’, ‘b = 2’, …, ‘z = 26’). The sequence follows a repeating pattern of step sizes between consecutive letters. The pattern is either a constant step (eg ‘+2, +2, +2, +2’) or a simple alternating pattern of two distinct steps (eg ‘+2, +3, +2, +3’).

Write a script to determine the letter that the '?' represents.

Examples

Example 1
Input: @seq = qw(a c ? g i)
Output: e
The pattern of the sequence is +2,+2,+2,+2.
1: a
3: c
5: e
7: g
9: i

Example 2
Input: @seq = qw(a d ? j m)
Output: g
The pattern of the sequence is +3,+3,+3,+3.
1: a
4: d
7: g
10: j
13: m

Example 3
Input: @seq = qw(a e ? m q)
Output: i
The pattern of the sequence is +4,+4,+4,+4.
1: a
5: e
9: i
13: m
17: q

Example 4
Input: @seq = qw(a c f ? k)
Output: h
The pattern of the sequence is +2,+3,+2,+3.
1: a
3: c
6: f
8: h
11: k

Example 5
Input: @seq = qw(b e g ? l)
Output: j
The pattern of the sequence is +3,+2,+3,+2.
2: b
5: e
7: g
10: j
12: l

Analysis

There are 5 positions (let's call them q0 to q4) of which one is initially a '?'. In the solved sequence we are told that if the pattern of the sequence is a, b, then:

q1 = q0 + a
q2 = q1 + b
q3 = q2 + a
q4 = q3 + b

As we know 4 out of the 5 q values we can determine a and b, because if the '?' is ...

q0, then b = q2 - q1 and a = q3 - q2
q1, then a = q3 - q2 and b = q4 - q3
q2, then a = q1 - q0 and b = q4 - q3
q3, then a = q1 - q0 and b = q2 - q1
q4, then a = q1 - q0 and a = q3 - q2

So, for example, if q2 is the '?', from the above we can see that
a = q1 - q0 and b = q4 - q3.

So now we know that if:

q0 is '?' then its letter is q1 - a
q1 is '?' then its letter is q0 + a
q2 is '?' then its letter is q1 + b
q3 is '?' then its letter is q2 + a
q4 is '?' then its letter is q3 + b

Try it 

#!/usr/bin/perl

# Blog: http://ccgi.campbellsmiths.force9.co.uk/challenge

use v5.26;    # The Weekly Challenge - 2026-04-27
use utf8;     # Week 371 - task 1 - Missing letter
use warnings; # Peter Campbell Smith
binmode STDOUT, ':utf8';
use Encode;

missing_letter(qw(? m b n c));
missing_letter(qw(b ? c b d));
missing_letter(qw(a c ? g i));
missing_letter(qw(a c e ? i));
missing_letter(qw(m a n b ?));
missing_letter(qw(z y x w ?));
missing_letter(qw(o o ? o o));

sub missing_letter {
    
    my (@q, $a, $b, $j, $qm, $gap, $char);
    
    # initialise
    @q = @_;
    
    # find the pairs (see blog)
    for $j (0 .. 3) {
        if ($q[$j] =~ m|[a-z]| and $q[$j + 1] =~ m|[a-z]|) {
            $gap = ord($q[$j + 1]) - ord($q[$j]);
            if ($j & 1) { $b = $gap } else { $a = $gap };
            
        # and the '?'
        } else {
            $qm = $j;
        }
    }
    $qm = 4 if $q[4] eq '?';
    
    # deduce the missing char
    $char = $qm == 0 ? chr(ord($q[1]) - $a) : chr(ord($q[$qm - 1]) + ($qm & 1 ? $a : $b));

    say qq[\nInput:  ] . join(' ', @q); 
    say qq[Output: ? = $char; sequence = ] . sprintf('%+d, %+d; ', $a, $b);
}

13 lines of code

Output

Input:  ? m b n c
Output: ? = a; sequence = +12, -11; 

Input:  b ? c b d
Output: ? = a; sequence = -1, +2; 

Input:  a c ? g i
Output: ? = e; sequence = +2, +2; 

Input:  a c e ? i
Output: ? = g; sequence = +2, +2; 

Input:  m a n b ?
Output: ? = o; sequence = -12, +13; 

Input:  z y x w ?
Output: ? = v; sequence = -1, -1; 

Input:  o o ? o o
Output: ? = o; sequence = +0, +0;