Arranging Letters and Digits

1. Write the word in capitals with a little space between the letters. 2. Under it, write the same letters in alphabetical order. Keep every repeat: DEVELOP has two Es, so the sorted row has two Es. 3. Compare column by column and count the columns where the top and bottom letters match. 4. 'How many change' is the word's length minus the number that stay.

• Fast route: a letter keeps its place only if exactly the right number of letters come before it in the alphabet. The 1st letter stays only if it is the earliest letter of the word; the last stays only if it is the latest. Checking the two ends first often settles the count.

1. Sort the letters into alphabetical order, keeping repeats. 2. Number the sorted row from the left. In a row of n letters, the kth from the right is the (n minus k + 1)th from the left. 3. Read off the two letters asked for. 4. Turn both into alphabet positions and count the letters between them: the difference minus 1.

• Fast route: you need only the two picked letters. The 2nd from the left is the 2nd earliest letter of the word, so you can often find it without writing the full row. This breaks when the word has repeats near the picked place: then write the row.

• Arranging digits: write the digits in ascending or descending order, keeping repeats, then read the places as you would for letters.
• A rule on each digit (add 1 to every even digit, take 1 from every odd one): change every digit first, then arrange. Doing it the other way round gives a different row.
• Digit facts: 0 + 1 + ... + 9 = 45. Any product that includes 0 is 0.
• Digital root: add the digits, and keep adding until one digit is left. It equals the remainder when the number is divided by 9, with 9 in place of 0.
• Palindrome: a number that reads the same backwards, such as 345543.
• Odd one out by a rule: test each option against the rule; the one that fails is the answer.