Why a "think of a number" trick always works, and how to invent one

What to assume they know

• Writing an unknown quantity as a letter and building an expression from it step by step — the Class 7 work the chapter's opening section says it is continuing
• Multiplying an expression by a whole number, and adding a number to it
• Dividing an expression by a whole number when every part of it divides cleanly
• Collecting like parts, so that a letter added and then taken away leaves nothing
• Solving a one-step equation of the form "something times the unknown equals a number"
• That the digits of a whole number carry different weights, so 126 holds a 1 in the hundreds and a 26 in the last two places (Part I, printed Chapter 5 uses this throughout)

What they should be able to do

• Follow a stated chain of arithmetic instructions with a letter in place of the starting number, writing the expression after each instruction
• Explain why a predicted answer is forced, by pointing at the step where the unknown's multiplier becomes nothing
• Alter one step of a trick so that it lands on a stated answer instead, and say which step controls the answer and why
• Invent a longer chain of instructions that always ends on the same value, and state the condition such a chain has to satisfy
• Track two unknowns — a month and a day — through six instructions and write the final expression
• Explain why the reported total can be decoded, using the fact that one unknown has been multiplied by a hundred and the other is small enough to fit in the last two places
• Recover a date from a reported total by subtracting the accumulated constant
• Redesign the date trick with different steps and work out the new constant that has to be subtracted

Where it usually goes wrong

• "It works because I tried it and it worked." Three successes are three successes. The expression covers every starting number at once, and that is a different kind of statement — this is the distinction the chapter's SUMMARY (Part II p.147) calls algebra's role in justification.
• "Halving undoes the doubling, so the +4 goes away too." It does not go away; it becomes +2. Students who treat division as an eraser get a prediction of 0 and then decide the trick is broken.
• "The prediction is 2 because the trick says so." The prediction is 2 because 4 was the number added before the halving. Change that one number and the prediction moves with it — which is exactly what the page asks for next.
• "The final subtraction is what makes the answer constant." The subtraction is the last of several moves; what matters is the total effect on the unknown. A chain that ends with the same subtraction but doubles without halving leaves the unknown standing, and predicts nothing.
• "The date trick works the same way as the first one." It works the opposite way. The first trick destroys the information; the date trick preserves it, in separate decimal slots. Both are read off the same kind of expression.
• "Any change to the steps keeps 165." 165 is built out of the particular additions and the multiplications that follow them. Move the 9 earlier and it gets multiplied twice; the number to subtract changes.
• "The trick would work whatever the day was." It needs the day to fit inside the last two places. A day of 31 is fine; a quantity of 150 would spill into the hundreds and the decode would name the wrong month.
• "The performer must be doing mental arithmetic very fast." The performer does one subtraction. Everything else was decided when the steps were designed.

Questions to check understanding

• Given a chain of instructions, write the expression after each step and state the prediction
• Given a required prediction, modify one step of a given chain to produce it, and justify the choice
• Decide whether a stated chain predicts a constant, and if it does not, say what the answer depends on
• Invent a chain of at least five instructions with a stated constant answer
• Given a reported total from the date trick, recover the month and the day
• Given a modified set of instructions for the date trick, state the number that must now be subtracted
• Explain, in terms of the size of the day, why the decoding step is unambiguous
• Short-answer reasoning: why does testing several starting numbers not prove a trick always works

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated inputs. No answer to any exercise the chapter sets the reader is printed anywhere in Part II pp.135–147, and Part II has no answer-key appendix. (The chapter does print worked answers to its own demonstrations — the 291 decode resolved to 25th December on Part II p.137, the apex-10 and apex-60 pyramids shown completed on Part II pp.138–139, the sum-36 block given as 5, 6, 12, 13 on Part II p.141, and the worked algebra grid resolving to 9 and 5 on Part II p.142 — so do not say the chapter prints no answers at all.)

• The opening chain of five instructions (Part II §6.2, p.135). In order: pick a secret number; take twice that; put four on top; halve what you now have; and finally remove the number you began with. The page states the prediction as 2 and invites the reader to try several starting numbers.
• The same chain with a letter (Part II §6.2, p.135). The page prints the running expression at every stage: the doubling gives twice the letter, adding four gives twice the letter plus four, halving gives the letter plus two, and the final subtraction leaves 2. The whole argument is in the halving step, because it turns the added 4 into an added 2 while cancelling the doubling.
• Redesigning the prediction (Part II §6.2, p.136 asks for 3 and for 5, and prints no answers). Verified: only the addition step needs changing — add 6 for a prediction of 3, add 10 for a prediction of 5. In general the addition has to be twice the answer you want, because the halving step is what the added number has to survive.
• A longer chain added here, for the "more complicated steps" prompt (Part II §6.2, p.136). Verified: pick a secret number, treble it, put twelve on top, divide by three, then remove the number you began with — the prediction is 4. The condition is the only thing worth saying: the unknown's multiplier must reach nothing by the last step, and then the leftover number is the prediction.
• The date trick as performed (Part II §6.2, p.136, a two-character comic strip; Shubham gives the instructions, Mukta computes). Six instructions, in order: five times the month; then six more; then four times that; then nine more; then five times again; and last of all the day joined on. Mukta's worked column for 26/01 is printed step by step: 1 × 5 = 5, 5 + 6 = 11, 11 × 4 = 44, 44 + 9 = 53, 53 × 5 = 265, 265 + 26 = 291. Shubham names the date as Republic Day.
• The same six instructions with letters (Part II §6.2, pp.136–137). The page prints the running expression: five times the month; five times the month plus 6; twenty times the month plus 24; twenty times the month plus 33; a hundred times the month plus 165; and finally a hundred times the month plus 165 plus the day.
• Where the two numbers 100 and 165 come from. Verified: the three multiplications are 5, 4 and 5, and 5 × 4 × 5 = 100, which is why the month ends up in the hundreds. The 6 is multiplied by the 4 and then by the 5, giving 120; the 9 is multiplied only by the 5, giving 45; and 120 + 45 = 165. The chapter prints 165 without decomposing it, and the decomposition is the whole content of section 12.
• The decode of Mukta's 291 (Part II §6.2, p.137). The page works it: 291 less 165 is 126, and since the day is at most 31 it occupies only the last two places, so the 1 in front is the month and 26 is the day.
• The chapter's second decode (Part II §6.2, p.137). A reported total of 1390; the page states that 165 off it gives 1225 and that the date was the 25th of December.
• Three totals set for the student (Part II §6.2, p.137): (i) 1269, (ii) 394, (iii) 296. Verified: 1269 − 165 = 1104, giving the 4th of November; 394 − 165 = 229, giving the 29th of February; 296 − 165 = 131, giving the 31st of January. Note when explaining it — the second of these is a leap-day, and the trick has no way of knowing whether the year had one. It decodes a month-and-day pair, not a date that necessarily exists.
• The two Math Talk prompts on Part II p.137, both in a printed marginal badge: change the steps and still be able to recover the date, in which case the number you subtract will not be 165; and devise a trick of your own. Verified worked answer for the first: cut the chain to four instructions — ten times the month, three more, ten times that, and the day joined on. The month is still carried up by a factor of a hundred, the additions accumulate to 30, and 30 is what you subtract.
• The chapter's own framing (Part II §6.1, p.135). The section says the two preceding years built the machinery and this chapter turns it on tricks and puzzles, including inventing new ones. That sentence is the reason section 12 exists and should not be dropped.

Figures to have open

• A two-column instruction table: the instruction in words on the left, the running expression on the right, one row per step. Needed twice — once for the five-step trick, once for the six-step date trick. Standard schematic, and the single most important figure in the explanation.
• A place-value strip for a four-digit total, showing the block that carries the month and the two places that carry the day, with the boundary drawn. Standard schematic; this is what makes section 9 land.
• A small decomposition diagram for 165: the 6 travelling through a ×4 and a ×5, the 9 travelling through a ×5, the two results meeting. Standard schematic; the chapter does not draw it.
• The comic strip on Part II p.136 need not be reproduced. Two speaker positions with the instruction on one side and the arithmetic on the other carry the same information; redraw rather than copy.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 6, "Algebra Play", §6.1 "Algebra Play" (Part II p.135) and §6.2 "Thinking about 'Think of a Number' Tricks" (Part II pp.135–137).
• The date trick's comic strip occupies most of Part II p.136; its algebraic reading runs from the foot of Part II p.136 to the middle of Part II p.137.
• Two Math Talk badges sit in the right margin of Part II p.137, one against the "change the steps" prompt and one against the "devise your own" prompt. The badge lettering is artwork and does not appear in extracted text.
• The chapter's SUMMARY (Part II p.147) states algebra's two roles — modelling and justification — and lists this section first among the things the chapter applied it to.
• The QR code on the chapter's opening page (Part II p.135) is captioned with the print code 0889CH06.
• Backward pointer: the chapter's opening sentence credits Class 7 for the original "Think of a Number" material and the last two years for the equation solving.