The alphabets from A to J are numbered from 0 to 9 respectively. Which one of the following is the value of AGJ – CEG + EDB?

Correct — A, CFE. The letters run A to J against the digits 0 to 9, so A is 0, B is 1, C is 2, D is 3, E is 4, F is 5, G is 6, H is 7, I is 8 and J is 9. Translate each three-letter block into the number it stands for. AGJ becomes 0, 6, 9, that is 69. CEG becomes 2, 4, 6, that is 246. EDB becomes 4, 3, 1, that is 431. Now do the arithmetic in the order written: 69 − 246 = −177, and −177 + 431 = 254. Turn 254 back into letters, digit by digit: 2 is C, 5 is F and 4 is E, giving CFE. The leading A in AGJ is the place where candidates slip, because A stands for zero and the block is worth 69, not 690 or anything larger.

• (b)DGF — DGF decodes to 3, 6, 5, that is 365 — a hundred and eleven more than the correct total. No misreading of the three blocks in the stem produces it.
• (c)GFD — GFD decodes to 653, which is the digits of the previous option reversed. It is what an answer would look like if the subtraction and addition were performed on reversed blocks.
• (d)FCE — FCE decodes to 524. This is the trap for reading AGJ as 690 by treating the leading A as a digit that occupies a place without being zero, which pushes the total up by 270 and then loses the hundreds digit.

A substitution code of this kind is a one-to-one map between symbols and digits, and the only reliable method is to convert everything into digits first, do ordinary arithmetic, and convert back at the end. Trying to add or subtract the letters directly is how errors creep in. The map here is the simplest possible one — the alphabet in order against the digits in order — which means each letter's value is its position in the alphabet minus one.

The one feature that makes this harder than it looks is the zero. A is worth nothing, so a block beginning with A is really a two-digit number wearing three letters, and the same is true of the answer if it had begun with A. Working the arithmetic in the printed order rather than rearranging it is the other discipline: 69 minus 246 is negative, and a candidate who avoids the negative intermediate by computing 246 minus 69 first will reach 608 rather than 254.

• The map is A = 0, B = 1, C = 2, D = 3, E = 4, F = 5, G = 6, H = 7, I = 8, J = 9.
• AGJ = 069 = 69, CEG = 246, EDB = 431.
• 69 − 246 + 431 = 254.
• 254 converts back as 2 = C, 5 = F, 4 = E, giving CFE.
• A leading A stands for zero, so a three-letter block beginning with A is worth only two digits.

• Reading A as 1 rather than 0, which shifts every value by one.
• Treating AGJ as a three-digit number of the order of six hundred.
• Rearranging the expression to avoid a negative intermediate and changing its value.

Asked as a decode-compute-encode chain where the whole item hinges on one letter standing for zero.

• practice — not a real PYQ

  If the letters A to J stand for the digits 0 to 9 in order, what does the sum BCD + EFG equal?

  • (a)FHJ
  • (b)FGH
  • (c)EGI
  • (d)GHI

  Answer(a) FHJ — BCD is 123 and EFG is 456, and 123 + 456 = 579, which reads back as F, H, J.

• practice — not a real PYQ

  If A to J represent 0 to 9 respectively, then the number represented by ACE is

  • (a)135
  • (b)24
  • (c)246
  • (d)13

  Answer(b) 24 — A is zero, so ACE is 0, 2, 4, which is the number 24.