NCERT Solutions Ganita Prakash (Part 1) Chapter 7 Puzzle Time
Updated on2026-09-05
Q1.
Solve the following Binairo puzzles: (first grid — a vertical line in row 2 column 3; horizontal lines in row 2 columns 5 and 6; a horizontal line in row 3 column 2; vertical lines in row 4 columns 3 and 5; horizontal lines in row 5 columns 1 and 5; a vertical line in row 6 column 3)
Answer
Here is the completed grid. Shaded cells are the ones printed in the book.
Puzzle 1 solved. The shaded cells are the ones the book gives; every row and every column has three vertical and three horizontal lines, no three alike are adjacent, and all six rows and all six columns are different.
The chain of reasoning, using V for a vertical line and H for a horizontal line:
Row 2 already has H at columns 5 and 6, so column 4 cannot be H — three in a row is not allowed. Column 4 is V. Now columns 3 and 4 are both V, so column 2 cannot be V; column 2 is H and column 1 is V. Row 2 is V H V V H H.
Column 3 now has V in rows 2, 4 and 6 — its full quota of three. Rows 1, 3 and 5 of column 3 are H.
Row 5 then has H at columns 1, 3 and 5 — three already — so columns 2, 4 and 6 are V: H V H V H V.
Column 1 has V in rows 2 and 3, so rows 1 and 4 must be H; that leaves row 6 as V to make three of each.
Carrying on the same way — count to three, never allow three alike side by side, and keep every row and every column different — fills the rest with no choice left anywhere.
Why the solution is unique: Every step above was forced, never guessed. Rule 1 (three of each in a line) and rule 2 (no three alike adjacent) together mean that as soon as two identical symbols sit side by side, the cells at both ends are decided. Rule 3 then removes the last ambiguity between two rows that would otherwise be interchangeable.
Q2.
Solve the following Binairo puzzles: (second grid — horizontal lines in row 1 column 1, row 2 columns 1 and 6, row 5 column 6; vertical lines in row 3 column 4, row 4 columns 2 and 3, row 6 columns 2 and 5)
Answer
The completed grid:
Puzzle 2 solved. The shaded cells are the ones the book gives; every row and every column has three vertical and three horizontal lines, no three alike are adjacent, and all six rows and all six columns are different.
Row 1: H H V V H V Row 2: H V V H V H Row 3: V H H V H V Row 4: H V V H H V Row 5: V H H V V H Row 6: V V H H V H
Column 1 begins with H in rows 1 and 2, so row 3 must be V.
Row 4 has V at columns 2 and 3, so columns 1 and 4 cannot be V — both are H.
Row 6 has V at columns 2 and 5; filling in the rest so that no three alike touch, and so that row 6 differs from every other row, leaves only V V H H V H.
Each row and each column ends with exactly three vertical and three horizontal lines, and all six rows and all six columns are different from one another.
Check it yourself: Read down each column — H H V H V V, H V H V H V, V V H V H H, V H V H V H, H V H H V V, V H V V H H. Six different columns, three of each symbol in every one, and nowhere three alike in a row.
Q3.
Solve the following Binairo puzzles: (third grid — a horizontal line in row 1 column 5, row 2 column 6, row 3 columns 1, 3 and 6; vertical lines in row 5 columns 3 and 4, and row 6 column 1)
Answer
The completed grid:
Puzzle 3 solved. The shaded cells are the ones the book gives; every row and every column has three vertical and three horizontal lines, no three alike are adjacent, and all six rows and all six columns are different.
Row 1: H V H V H V Row 2: V H V H V H Row 3: H V H V V H Row 4: V H V H H V Row 5: H H V V H V Row 6: V V H H V H
Row 3 is given H at columns 1, 3 and 6 — that is all three of its horizontal lines, so columns 2, 4 and 5 must be V.
Row 5 has V at columns 3 and 4, so column 2 and column 5 cannot be V; that starts the row as H H V V H, and the count of three forces column 6 to be V.
Working down each column and keeping the three-of-each count gives the rest with no choices left.
Why counting beats guessing: In a 6 by 6 Binairo every line holds exactly three of each symbol. The instant three of one kind are placed in a line, every remaining cell in it is settled. Looking for the line that is closest to its quota — as row 3 was here — is the fastest way in, and it is the same habit that makes a proportion easy: find the term you can pin down first.