NCERT Solutions for Class 7th Maths Chapter 2 A Magic Grid of Integers — Multiplication of Integers

Book page 37–38 Updated on2026-09-19

Q1.

A grid containing some numbers is given below. Follow the steps as shown until no number is left. Circle any number; strike out the row and the column containing that number; circle any unstruck number; when there are no more unstruck numbers, stop. Multiply the circled numbers.

Answer

The product is –30240, and the book’s own example confirms it.

8	–4	12	–6
–28	14	–42	21
12	–6	18	–9
20	–10	30	–15

In the four rounds shown in the book the circled numbers are –6, 14, 20 and 18.

(–6) × 14 = –84
(–84) × 20 = –1680
(–1680) × 18 = –30240

Notice that the four circles use each row once and each column once — the striking-out rule forces that.

Why it happens: the very first circle removes its own row and column from play, so the next circle must come from a different row and a different column, and so on. With a 4 × 4 grid you make exactly 4 circles, one in each row and one in each column.

Q2.

Try again, and choose different numbers this time. What product did you get? Was it different from the first time? Try a few more times with different numbers!

Answer

The product is –30240 again — every single time.

Try three completely different sets of circles.

Circled numbers	Working	Product
8, –42, –9, –10	8 × (–42) = –336; × (–9) = 3024; × (–10)	–30240
–4, –28, 18, –15	(–4) × (–28) = 112; × 18 = 2016; × (–15)	–30240
–6, –42, –6, 20	(–6) × (–42) = 252; × (–6) = –1512; × 20	–30240

The reason is hidden in how the grid was built. Every entry is a row number times a column number:

×	4	–2	6	–3
2	8	–4	12	–6
–7	–28	14	–42	21
3	12	–6	18	–9
5	20	–10	30	–15

Each circle contributes one row number and one column number
All four row numbers get used: 2 × (–7) × 3 × 5 = –210
All four column numbers get used: 4 × (–2) × 6 × (–3) = 144
Product = (–210) × 144 = –30240

Why it happens: because multiplication is commutative and associative, the sixteen factors may be rearranged freely. Whichever cells you circle, you pick up each row number exactly once and each column number exactly once — so the same eight numbers are always multiplied, only in a different order. The answer therefore cannot change.

Q3.

Play the same game with the grid below. What answer do you get?

8	–4	12	–6
–28	14	–42	21
12	–6	18	–9
20	–10	30	–15

The grid printed on page 38 for the second round of the game.

Answer

The answer is –30240 once more.

In the copy of the book being used here the second grid carries exactly the same sixteen entries as the first — 8, –4, 12, –6 / –28, 14, –42, 21 / 12, –6, 18, –9 / 20, –10, 30, –15 — so its magic number is the same –30240.

Row numbers: 2, –7, 3, 5 → product –210
Column numbers: 4, –2, 6, –3 → product 144
Magic number = (–210) × 144 = –30240

Check it yourself: whatever grid your copy shows, you do not have to play the game to find its magic number. Read off the four row numbers and the four column numbers, multiply all eight together, and you have the answer.

Why it happens: the game is really just a disguised way of multiplying the four row numbers by the four column numbers. The circling only decides the order of the multiplication, and order never affects a product.

Q4.

What is so special about these grids? Is the magic in the numbers or the way they are arranged or both? Can you make more such grids?

Answer

The magic is in the arrangement — the numbers are ordinary, but they are laid out as a multiplication table.

Every cell is (its row number) × (its column number). Because the rules force one circle per row and one per column, the final product is always

(r₁ × r₂ × r₃ × r₄) × (c₁ × c₂ × c₃ × c₄)

Making your own grid. Pick any four row numbers and any four column numbers, then fill each cell with their product. Here is a fresh one built from rows 1, –2, 4, 3 and columns 5, –1, 2, –4.

×	5	–1	2	–4
1	5	–1	2	–4
–2	–10	2	–4	8
4	20	–4	8	–16
3	15	–3	6	–12

Rows: 1 × (–2) × 4 × 3 = –24
Columns: 5 × (–1) × 2 × (–4) = 40
Magic number = (–24) × 40 = –960

Rub out the header row and column before you show it to a friend, and the grid looks like sixteen unrelated integers.

Why it happens: the trick rests entirely on commutativity and associativity. Those two properties let the eight hidden factors be gathered in any order, so the product is fixed before the game even begins.

Tip: to make the answer positive, use an even number of negatives among your eight chosen numbers. To make it 0, put a 0 in the list — then every game ends at 0.

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