Cracking a cryptarithm by reasoning about digits, not guessing

What to assume they know

• Place value, and reading a multi-digit number as a sum of its place values (Powers of 10, and place value written out for whole numbers and decimals)
• Column addition with carrying, and column multiplication by a one-digit number
• Multiplication tables to ten, and the units digits of products
• Comparing sizes: knowing how many digits a product can have
• Cryptarithms as met in Class 7, which this section explicitly picks up from
• Counterexamples and elimination (Always, sometimes, or never: one counterexample settles it)

What they should be able to do

• State the three rules a cryptarithm obeys and explain what each one rules out
• Work a written addition column by column, saying what each column determines
• Reason about the amount carried from one column to the next, and bound it
• Use the number of digits in the answer to fix the first digit of a factor
• Use the units digit of a product to eliminate candidate digits
• Reduce a multiplication cryptarithm to a small list of possibilities and then to a solution, justifying every elimination
• Check a completed solution against all three rules and against the arithmetic
• Decide whether a cryptarithm has more than one solution, and say how you know

Where it usually goes wrong

• "You solve these by trying combinations." Trying is what the method replaces. Every printed passage in this section moves by consequence: a size bound, a units digit, a rule violation. Model the reasoning out loud, and when the explanation does narrow to a shortlist, say what narrowed it.
• "Every digit must have a letter." The rule runs the other way: no digit may have two letters. Bare digits sit openly inside several of these puzzles, and a student who thinks each of the ten digits needs a letter will read the three-digit puzzle with a 2 in it as broken.
• "The same letter could be a different digit in a different place." It could not. A letter is one digit everywhere it appears, which is exactly why a repeated letter — the two Ps, the three Gs, the W appearing twice — is the most informative thing in a puzzle.
• "Two different letters might happen to be the same digit." Forbidden, and Guna uses that ban directly to eliminate a candidate.
• "A number can start with zero if the arithmetic works." It cannot, and this rule quietly does a lot of work at the top of every puzzle.
• "The amount carried is always one." Adding two digits can pass at most one; adding three can pass two; multiplying can pass considerably more. Bound the carry from the actual operation rather than from habit.
• "Finding one solution finishes the job." Some of these puzzles have more than one, and a solver who stops at the first has answered a different question. Ask for the argument that no others exist.

Questions to check understanding

• Solve an addition cryptarithm and justify each digit as it is fixed
• Solve a multiplication cryptarithm by a one-digit multiplier
• Use a size argument to fix the first digit of a factor, given the digit count of the product
• Eliminate candidate digits using the units digit of a product
• Explain which rule a proposed solution breaks
• Say whether a cryptarithm has a unique solution, and argue the case
• Construct a cryptarithm of your own with a stated solution, and check that the rules hold

Examples worth working on the board

Inputs. Values marked "printed" are the chapter's own reasoning; everything else is a puzzle left open.

• The three rules, printed (Part I, §5.3, p.131, opening lines). One letter means one digit wherever it appears; no two letters may land on the same digit; and no number in the puzzle opens with a zero. The chapter also says it is picking up cryptarithms from the previous year, so the explanation can assume a first acquaintance. Note the exact shape of the second rule: it forbids two letters sharing a digit, and it does not require every digit to have a letter — which is why bare digits appear inside several of these puzzles.
• The four addition puzzles, printed (Part I, §5.3, p.131, set out as column sums): (i) a two-digit number written A then 1, added to a two-digit number written 1 then B, giving a two-digit result written B then 0. (ii) a two-digit number written A then B, added to 37, giving a two-digit result written 6 then A. (iii) a two-digit number written O then N, taken three times, giving a two-digit result written P then O. (iv) a two-digit number written Q then R, taken three times, giving a three-digit result written P, R, R. Items (iii) and (iv) are printed as three-line column sums, not as products.
• The first multiplication puzzle and Guna's reasoning, printed (Part I, §5.3, p.131). The puzzle is a two-digit number written P then Q, multiplied by 8, giving a two-digit result written R then S. Guna's argument, as printed: a two-digit number times eight must still be two digits; 10 × 8 = 80, but that makes the units digits of the two numbers match, which the rules forbid; the number cannot be 11 either, since its two letters would be the same digit; and 12 × 8 = 96 satisfies everything. The page then asks whether 13 could work and answers itself — 13 × 8 = 104, and every two-digit number above 12 gives a three-digit product. A drawing of Guna, a boy in a yellow shirt with his arms spread, sits beside the passage.
• The second multiplication puzzle, printed (Part I, §5.3, p.132). A two-digit number written G then H, multiplied by H, giving a two-digit result written 9 then K. The chapter says the product lies in the nineties and tells the student to look at which letters occupy the units positions, then offers seven candidate products to choose from: 11 × 9 = 99, 12 × 8 = 96, 46 × 2 = 92, 24 × 4 = 96, 47 × 2 = 94, 31 × 3 = 93, and 16 × 6 = 96. Two conditions do the eliminating — the multiplier is the same letter as the units digit of the two-digit number, and no two letters may carry the same digit.
• The three-letter puzzle and Anshu's reasoning, printed (Part I, §5.3, p.132). A three-digit number written B, Y, E, multiplied by 6, giving a three-digit result written R, A, Y. Anshu's argument, as printed: the product has three digits, so B cannot be 2 or more, because two hundreds multiplied by six already comes to at least 1200; therefore B is 1. The page then asks what can be said about Y, and Anshu answers that Y cannot be 7 or more, since 170 × 6 = 1020 and a three-digit product is wanted, and adds that Y must be even. A drawing of Anshu, in a green checked shirt with one finger raised, sits beside the passage. Note that the reason Y must be even is not given on the page; supplying it is one of the explanation's jobs.
• The six puzzles set for the student, printed (Part I, §5.3, p.132). In order: a two-digit number written U then T, times 3, giving a three-digit result written P, U, T; a two-digit number written A then B, times 5, giving a two-digit result written B then C; a three-digit number written L, 2, N, times 2, giving a three-digit result written 2, N, P; a two-digit number written X then Y, times 4, giving a two-digit result written Z then X; a two-digit number written P then P, times a two-digit number written Q then Q, giving a three-digit result written P, R, P; and a two-digit number written J then K, times 6, giving a three-digit result written K, K, K. The third of these carries a bare digit 2 inside it alongside its letters — a clean demonstration of what the second rule does and does not say.
• Two more at the chapter end, printed (Part I p.133 no. 15): a two-digit number written E then F, multiplied by E, giving a three-digit result written G, G, G; and a three-digit number written W, O, W, multiplied by 5, giving a four-digit result written M, E, O, W.
• The chapter prints a complete solution for none of these. Guna's and Anshu's passages are partial reasoning, not answers.

Figures to have open

• A column-sum frame with separable columns and a visible carry slot, reusable across all the addition puzzles. Standard schematic; the chapter prints its puzzles as plain column sums (Part I, §5.3, p.131) and the explanation needs the structure exposed.
• A letter-to-digit assignment board: one slot per letter, filled as deductions land, with a clash indicator when two letters reach for the same digit. Standard schematic. This is the object that makes the three rules operational.
• A product-size number line: two-digit numbers along an axis, with the point marked where multiplying by the given factor pushes the product to three digits. Standard schematic; it carries both Guna's argument and Anshu's.
• The seven candidate products from Part I p.132 as a strike-through list.
• The two character drawings on Part I pp.131–132 are decorative; the reasoning in the speech beside them is what matters, and a plain speech bubble carries it.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 5, "Number Play", §5.3 "Digits in Disguise", Part I pp.131–132. The section opens at the foot of Part I p.131, immediately after the digital-roots exercises, and its final set of puzzles is on Part I p.132.
• Chapter-end "Figure it Out", Part I p.133, item 15.
• Backward pointer: the chapter states that cryptarithms were met in the previous year's course, so this is a return rather than an introduction.
• The chapter's SUMMARY, Part I p.134, does not mention cryptarithms; §5.3 is treated as an application of the chapter's reasoning habits rather than as a new result.
• The spine's other §5.3-adjacent topic, the chapter-end board game, is the chapter-end board game on Part I p.135 and is marked video: no.