Constructing magic squares, and where they came from

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

Using row and column sums to prove a grid cannot be filled: the row and column totals of a 3 × 3 grid filled with 1 to 9 must each come to 45, and each single total is trapped between 6 and 24
Adding three one-digit and three two-digit numbers reliably
Brackets decide which operation happens first and the calendar-grid work in the Arithmetic Expressions chapter: describing several cells as one letter-number plus or minus something
Consecutive numbers, and what adding a fixed amount to a whole set does
Reading a claim of the form "this cannot happen" as something that needs a reason

What they should be able to do

State the defining condition of a magic square, including the diagonals, and name the magic sum
Show that a magic square built from 1 to 9 must have magic sum 15
Show that its centre must be 5, using the chapter's elimination argument
Explain why the smallest and largest numbers cannot sit in a corner
Complete a 3 × 3 magic square once one full line has been placed
Predict what happens to a magic square, and to its magic sum, when every entry is increased by a fixed amount or multiplied by a fixed amount
Write a 3 × 3 magic square of consecutive numbers as offsets from its middle entry, and use that form to build one to order
Name the Chautīsā Yantra and the Lo Shu Square, say roughly when and where each is from, and read the magic sum off each

Where it usually goes wrong

"Magic just means the rows and columns match." The diagonals are part of the definition, and they are what makes the centre special. A grid with matching rows and columns and mismatched diagonals is the standard near-miss; show one.
"You find a magic square by trying arrangements until one works." With 3,62,880 arrangements that is not a method, it is a hope. Every step from Observation 1 onward cuts the search down by reasoning, and by the end there is nothing left to search.
"Nine cannot be at the centre because it is the biggest." Being biggest is not the reason. The reason is that every other cell shares a line with the centre, so a 9 in the middle would have to make 15 with the 8 as well — and 8 and 9 already overshoot. Say the shared-line fact out loud; without it the elimination looks like a lucky choice of example.
"There are lots of different 3 × 3 magic squares using 1 to 9." There is one, appearing in eight guises — the four rotations and their mirror images. The chapter poses the counting question and leaves it open.
"Adding 1 to every entry adds 1 to the magic sum." It adds 3, because each line has three cells. This is the single most common slip on this page, and the offsets form makes it obvious.
"Magic squares need consecutive numbers." They do not — the chapter sets exactly that question, and multiplying every entry of a magic square by a constant already breaks consecutiveness while keeping the square magic.
"The Chautīsā Yantra is a curiosity from a museum." It is a dated tenth-century inscription in a working temple, and 3 × 3 squares of the same family are still painted and sold today. Sections 11 and 12 exist to make the mathematics continuous with that, not to decorate it.

Questions to check understanding

State the magic sum of a 3 × 3 magic square built from 1 to 9, with a reason
Complete a magic square from two or three given entries
Explain why the centre entry is forced, and why 1 and 9 avoid the corners
Build a magic square to a stated centre, or to a stated magic sum, including a magic sum of 0 with negative entries (question 4 on p.144)
Say what happens to a magic square and its magic sum under adding a constant or multiplying by a constant
Decide whether a given filled grid is magic, with the diagonals checked
Read the magic sum off an unfamiliar magic square, such as one of the yantra panels, and say which nine numbers it uses
Date-and-place questions on the Chautīsā Yantra and the Lo Shu Square appear as short-answer history-of-mathematics items

Examples worth working on the board

3,62,880 (Part I, §6.3, p.134). Printed in the Indian grouping as the count of ways to fill a 3 × 3 grid with 1 to 9 without repeating. The chapter says plainly that this count can be found without listing them, and defers how to later classes — do not invent the method on its behalf.
The diagonals figure (Part I, §6.3, p.134). Checked against the printed page: a small 3 × 3 grid with the two diagonals drawn as thick bands crossing at the centre. This is the figure that stops a reader assuming rows and columns are the whole condition.
Observation 1 (Part I, §6.3, pp.134–135). Inputs: the three row totals of a magic square are equal, and together they come to 45.
The centre eliminations (Part I, §6.3, p.135). Checked against the printed page. Two small grids are printed alongside the argument: one with 8 in the top-left corner and 9 at the centre, a dashed diagonal joining them; one with 1 at the centre and 2 below it, a dashed line down the middle column. Inputs: a centre of 9 with an 8 elsewhere, and a centre of 1 with a 2 elsewhere. In each case the third number on the shared line is impossible.
Observation 2 (Part I, §6.3, p.135). The centre must be 5, with a printed grid showing 5 alone in the middle cell.
The two position types (Part I, §6.3, p.135). Checked against the printed page: two blank 3 × 3 grids side by side, the first with dots in the four corners, the second with dots in the four edge cells. This is the chapter's own classification of the boundary and it is what section 6 hangs on.
The corner argument (Part I, §6.3, p.136). Checked against the printed page: a grid with 1 in the top-left corner and 5 at the centre, with 15 written against the row, the column and the diagonal through that corner. Inputs: 1 + 5 + 9 = 15 and 1 + 6 + 8 = 15, and the question of whether a third such combination exists.
The two placements of 1, 5, 9 (Part I, §6.3, p.136). Checked against the printed page: one grid with 1, 5, 9 filling the middle row and one with 1, 5, 9 filling the middle column. From either, the square completes.
The generalised grid (Part I, "Generalising a 3 × 3 Magic Square", §6.3, pp.136–137). Checked against the printed page: a blank 3 × 3 grid with a single italic m printed in the centre cell and nothing else. The chapter asks the reader to express every other cell as an amount above or below m, and points back to the calendar grid in the Arithmetic Expressions chapter for the technique. It does not print the completed offsets — the explanation derives them.
Build-to-order tasks (Part I, §6.3, p.137). Inputs, exactly as set: a magic square whose centre is 25; a magic square with magic sum 60; and the question of whether nine non-consecutive numbers can be made to work.
The zero-sum square (Part I, §6.5 Figure it Out, p.144, question 4). The chapter's parting construction task, set in the chapter-end mixed exercise rather than inside §6.3: build a 3 × 3 square whose eight lines all total 0, negative entries allowed, and not every entry zero. It is the offsets form of section 10 read at m = 0, so an explanation that has done section 10 can answer it without a fresh idea — which is exactly why it makes the closing beat. One filling is 3, −4, 1 / −2, 0, 2 / −1, 4, −3; every row, every column and both diagonals come to 0, and the nine entries are the nine offsets −4 up to 4. The question fixes only the magic sum, not the nine numbers, so other fillings exist besides the eight turns and flips of this one — 5, −7, 2 / −3, 0, 3 / −2, 7, −5 is another, which I worked and checked. The chapter body prints no answer; the solutions appendix bound after the chapter gives the offsets square and says explicitly that it is only one of several.
The Chautīsā Yantra (Part I, §6.3, p.137). Checked against the printed page. A photograph of the inscribed stone panel beside a typeset copy reading 7 12 1 14 / 2 13 8 11 / 16 3 10 5 / 9 6 15 4. It was cut in the tenth century into the Jain temple of Pārśhvanath at Khajuraho. Its magic sum is 34, and the chapter glosses the name accordingly. It also asks the reader to hunt for other four-cell groupings reaching 34.
The Lo Shu Square (Part I, §6.3, p.138). Checked against the printed page. Printed as 2 7 6 / 9 5 1 / 4 3 8, beside a drawing of a turtle carrying a dotted grid on its shell. Over two thousand years old; ancient China; the flood on the Lo River and the turtle sent to save the people.
The Palani pillar (Part I, §6.3, p.138). A photograph of a magic square carved into a pillar at Palani, Tamil Nadu, whose temple is dated to the 8th century CE. The numerals in the photograph are worn and are not transcribed in the book — do not invent them.
The Navagraha Yantra (Part I, §6.3, p.138). Checked against the printed page. Nine coloured panels in a 3 × 3 arrangement, labelled Mercury, Venus, Moon, Jupiter, Sun, Mars, Ketu, Saturn and Rahu, each carrying its own small 3 × 3 magic square. The chapter's own comment is only that the magic sum differs from panel to panel. See Notes for what the panels actually contain, including one misprint.
The Kubera Yantra (Part I, §6.3, p.139). Checked against the printed page. A painted yantra with its 3 × 3 square drawn out beside it: 27 20 25 / 22 24 26 / 23 28 21. Nine consecutive numbers, middle entry 24.

Figures to have open

An editable 3 × 3 grid that can show its eight lines individually, hold partial fills, and switch between numerals and offsets from m.
A cell-type diagram: corner, edge and centre, each annotated with the number of lines running through it. This is the figure that carries the thesis.
A 4 × 4 grid for the Chautīsā Yantra, able to highlight arbitrary four-cell groupings.
A turtle with a dotted shell grid, for the Lo Shu legend — redraw; the printed turtle is the book's artwork.
The Khajuraho inscription and the Palani pillar are photographs of physical objects and must come from the textbook or from another cleared source; they cannot be schematised without losing the point that these are real inscriptions. If neither can be cleared, show the typeset grids and say when explaining it where the originals are.
A nine-panel yantra layout for section 12. Redraw the panels rather than lifting the printed art, and see Notes before transcribing the numbers into it.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 6 "Number Play", §6.3 "Some Explorations in Grids", pp.134–139 — the definition of a magic square and the count of arrangements (p.134), Observations 1 and 2 with the centre eliminations (p.135), the corner argument and Observation 3 with the Figure it Out set (p.136), the unnumbered block "Generalising a 3 × 3 Magic Square" (pp.136–137), the unnumbered block "The First-ever 4 × 4 Magic Square" (p.137), and the unnumbered block "Magic Squares in History and Culture" (pp.138–139)
Same part, §6.5 Figure it Out, p.144, question 4 — the zero-sum magic square, set in the mixed chapter-end exercise rather than inside §6.3, which is why it sits outside the page range above
Same part, SUMMARY, p.145, fourth bullet, for the chapter's own one-line account of extending the grid work into magic squares
Backward pointer: the calendar-grid generalisation in the Arithmetic Expressions chapter, which p.137 names as the technique to reuse — Part I, printed Chapter 2
Solutions appendix bound after p.145 in the cached PDF (not part of the printed book), consulted for the intended answers to the p.136 and p.137 question sets and to question 4 on p.144