Calendar magic: the position in the grid is the whole secret

What to assume they know

• Writing an unknown as a letter and building expressions such as "the letter plus seven"
• Collecting like parts, so that four copies of an unknown become four times it
• Solving a two-step equation: subtract from both sides, then divide both sides
• Substituting a found value back into expressions to recover the other quantities
• Reading a printed calendar month: seven columns, one row per week, ragged ends
• That a multiple of four leaves no remainder when divided by four

What they should be able to do

• State the two constant steps of a calendar grid and justify them from how the month is laid out
• Express every cell of a chosen block in terms of one chosen cell of it
• Add those expressions and simplify to a rule of the form "four times the corner plus a fixed number"
• Given a stated total, solve for the corner and write out all four dates
• Verify a recovered block against the printed calendar
• Decide whether a stated total is possible at all for a given block shape
• Say where a block of a given shape is not allowed to sit, and why the argument fails there
• Compute the fixed number for a new block shape, and so design a trick of your own
• Transfer the whole method to a grid with a different number of columns, and state what changes

Where it usually goes wrong

• "The trick needs a calendar." It needs a grid with two constant steps. The nine-column grid on the facing side of the same page is there to make exactly that point, and the chapter's prompt to invent your own trick is an invitation to break the calendar.
• "Seven appears because a week has seven days." True but not the reason that matters. Seven appears because the month is printed seven columns wide. Print the same dates ten to a row and the downward step becomes ten.
• "The 16 is something about August." It is not. It is the total of the block's own offsets — nothing, one, seven and eight — and it is the same in every month of every year, on any seven-column calendar.
• "Any total can be decoded." Only totals of the form four times a whole number plus 16. If a friend reports 38, either they added wrongly or the block was not 2 × 2.
• "The block can start anywhere." Straddling the right-hand edge breaks the step of one, because the next date is on the next row. This is the single most common failure when students try the trick on each other.
• "The total is four times the middle number." There is no middle cell in a 2 × 2 block. The 3 × 3 block does have one, and its total is nine times that centre — which is why the two cases must be set side by side rather than conflated.
• "You need all four dates to find the total, so you need the total to find all four." You need one date. The other three are consequences, and the total is a disguised way of handing you the one.
• "A different shape needs a different method." It needs a different offset. The method — write every cell in terms of one, add, solve — is untouched.

Questions to check understanding

• Write the cells of a stated block shape in terms of one chosen cell, and give the total
• Given a total and a block shape, recover every entry
• Decide whether a stated total is achievable for a stated block shape, with a reason
• Given a printed month, list the positions where a stated block cannot be placed
• Design a trick with a block shape of your own choosing and state its offset
• Repeat the whole argument on a grid with a stated number of columns
• Explain why a 3 × 3 total is nine times the centre entry but a 2 × 2 total is not four times any entry

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 printed calendar month (Part II §6.4, p.141). August 2025, seven columns headed Sunday to Saturday. Read off the printed page: the 1st falls on a Friday and the 2nd on a Saturday, the month runs to 31 days, and the 31st sits alone in the Sunday column of a sixth row. The first row is therefore mostly empty and the last row holds a single date — both facts matter for section 8.
• The block the page highlights (Part II §6.4, p.141). A 2 × 2 block holding 6, 7, 13 and 14, with its total printed as 40.
• The same block in letters (Part II §6.4, p.141). The chapter names the top-left cell and prints the other three as that cell plus one, plus seven and plus eight; it then prints the total as four times the cell plus 16.
• The recovery the chapter performs (Part II §6.4, p.141). Told the total is 36: subtract 16 from both sides to get four times the cell equal to 20, divide both sides by four to get 5, and the block is printed as 5, 6, 12, 13.
• Closing the loop the chapter leaves open. Verified: run the same recovery on the page's own printed total of 40 — subtract 16 to get 24, divide by four to get 6 — and the block comes back as 6, 7, 13, 14, which is the block the page started from. The chapter demonstrates the trick on one total and the recovery on a different one, so this check is worth doing.
• Which totals are reachable. Verified: since the total is four times the corner plus 16, it is always a multiple of four, and the smallest one August 2025 allows is 4 × 1 + 16 = 20, from the block 1, 2, 8, 9. A total of 38 is impossible; a total of 20 is not.
• Where the block may not sit. Verified against the printed month: the argument needs a cell to the right in the same row and a row below. In August 2025 the 9th is a Saturday, so the block that arithmetic would build from it — 9, 10, 16, 17 — is not a block of the calendar at all, because the 10th starts the next row. The same failure hits every Saturday. The whole of the last full row is barred as well, the 24th included — the 24th would need 31 and 32 beneath it and there is no 32. So in August 2025 (Fri 1 to Sun 31, rows {1, 2}, {3–9}, {10–16}, {17–23}, {24–30}, {31}) the legal top-left corners are exactly 1, 3–8, 10–15, 17–22, and what is barred is every Saturday — 2, 9, 16, 23, 30 — together with every date from 24 to 31. Section 8 shades the disallowed corners on the printed month, so getting the 24th onto the shaded side matters.
• A second calendar and a second grid (Part II §6.4, p.142, under the Math Talk prompt that asks for a trick of your own with a block of a different size and shape). The page reprints the August 2025 month with several regions outlined in red, of assorted sizes and shapes; the largest and clearest of them is a 3 × 3 block covering 10, 11, 12 / 17, 18, 19 / 24, 25, 26. Beside it is a separate nine-column grid whose rows run 2 to 10, 12 to 20, 22 to 30, 32 to 40 and 42 to 50, also carrying red-outlined regions. None of them is a full-height column strip, and the column holding 3, 13, 23, 33 and 43 is the place a redraw goes wrong: resolved on the printed page, that column is crossed by three separate regions of three different shapes — a vertical domino {3, 13}; a horizontal four-cell strip {22, 23, 24, 25}, whose red boundary runs above and below the whole of it and closes at the grid's left edge, so 23 belongs to the strip and not to any column; and a second vertical domino {33, 43}. Red ink does run the length of the column, which is what it looks like at lower resolution, but it is three regions meeting, not one. Further regions sit to the right: a domino on 18, 19; a vertical domino on 27, 37; a strip across 28, 29, 30 closing at the grid's right edge; and a strip on 46, 47, 48.
• Shape offsets for other blocks (all verified, none printed). On a seven-column calendar: three in a row gives three times the corner plus 3; three in a column gives three times the corner plus 21; an L of three — the corner, the cell below it and the cell to the right of that — gives three times the corner plus 15; a 3 × 3 block gives nine times the corner plus 72. That last one has a pleasant consequence worth showing: nine times the corner plus 72 is nine times the centre cell, so a 3 × 3 total is always nine times its middle date. Checked on the printed outline: the 3 × 3 block on Part II p.142 has 18 at its centre and totals 162.
• The 2 × 2 block has no such centre. Verified: four times the corner plus 16 factorises as 4(a + 4), and a + 4 is not one of the block's four dates — those are a, a + 1, a + 7 and a + 8. (Say it as the factorisation, not as "four times the corner plus four", which reads as 4a + 4 and is false.) This is a good contrast to draw immediately after the 3 × 3 result, because students will try to generalise the centre shortcut.
• The nine-column grid. Verified: in the grid printed on Part II p.142 one step right is one more and one step down is ten more, so a 2 × 2 block there totals four times its corner plus 22. Nothing else about the method changes.

Figures to have open

• The August 2025 month as printed (Part II p.141), with the right and down arrows overlaid. The chapter's own figure; redraw it as a clean grid rather than reproducing the printed art, but keep the real layout — the Friday start and the lone 31st are load-bearing for section 8.
• A 2 × 2 block with its cells labelled by offset from the corner, and the offsets totalled separately. Standard schematic; this is the figure the whole argument rests on.
• A table of block shapes against their offsets, for section 9. Standard schematic, and not in the chapter.
• The nine-column grid of Part II p.142 running 2 to 50, with its own arrows. Redraw; the chapter prints it with red outlines over several regions, and the outlines are not needed once the arrows are shown.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 6, "Algebra Play", §6.4 "Fun with Grids", under the printed unnumbered subheading "Calendar Magic", Part II pp.141–142. The subheading can be named but has no section number.
• The August 2025 calendar and the highlighted 2 × 2 block occupy the upper third of Part II p.141; the lettered block, the total and the two-step recovery fill the rest of that page.
• The invent-your-own prompt, the second calendar with its red-outlined regions and the nine-column number grid all sit in the top third of Part II p.142, with a Math Talk badge in the outer margin. The badge lettering is artwork and does not appear in extracted text.
• The Algebra Grids passage that follows on the same page (Part II p.142) is a different topic and is covered by Algebra grids: every row is an equation, and the shapes are the unknowns.
• The chapter's SUMMARY (Part II p.147) lists grids among the things algebra was applied to here.