Chapter focus
Number puzzles turn arithmetic into a search for structure. A magic square is one example: its rows, columns and diagonals share a common total.
NCERT CIET Grade 7 includes constructing and investigating a 4 × 4 magic square with magic constant 34. Use the full Ganita Prakash Part I activity for its prescribed exploration.
1. What makes a magic square?
A magic square is a square array of numbers in which each row, each column and the two main diagonals have the same sum, called the magic constant. For the numbers 1 through 16 arranged once in a 4 × 4 square, the total of all entries is 1 + 2 + … + 16 = 136. Since four rows share the same row total, the magic constant must be 136 ÷ 4 = 34.
This total gives a check, not a full construction. Each row, column and main diagonal still needs to be tested.
2. Use constraints to solve a number puzzle
In a magic square, entries are not independent. If three numbers in a row add to 25, the fourth must be 34 − 25 = 9. But that value must also satisfy the relevant column and diagonal totals and must not duplicate a number if the puzzle requires each number to be used once.
Keep a table of constraints: required row sum, column sum, diagonal sum, allowed numbers and used numbers. Cross out a candidate as soon as it violates one of these rules.
3. Check a candidate square systematically
- Add each row and verify it equals the target total.
- Add each column and verify the same total.
- Check both main diagonals.
- Check that every required entry appears the correct number of times and no forbidden number is used.
Do not check only the rows: an arrangement may satisfy all row sums while failing a column or diagonal.
4. Other useful number patterns
Look for sums that stay constant, numbers that are paired, or a repeated change between terms. For example, in 2, 5, 8, 11, … each term increases by 3. Explain the rule, test later terms and distinguish an observed pattern from a proof that it must always continue.
For a systematic search, list what is fixed and what can vary. Try a candidate, record why it succeeds or fails, and use the failure to narrow the next choice.
Practice: reason from the constraints
1. What is the magic constant for a 4 × 4 magic square using 1 through 16 once each?
Answer: The total is 136 and there are four rows, so each row must sum to 136 ÷ 4 = 34.
2. Three entries in a row sum to 26. What must the fourth be if the target is 34?
Answer: 34 − 26 = 8.
3. A 4 × 4 candidate has row sums 34, 34, 34 and 34. Is it necessarily a magic square?
Answer: No. Its columns and both main diagonals must also sum to 34, and the required entries must meet the puzzle’s conditions.
4. Continue 6, 10, 14, 18, … for three terms and state a rule.
Answer: 22, 26, 30; add 4 each time.
5. Why should you record which numbers have already been used in a puzzle?
Answer: It prevents accidental repetition when the task requires each number to appear once and helps narrow valid choices.
Chapter summary
Number puzzles are solved by identifying constraints and checking them systematically. For the 1–16 4 × 4 magic square, the row total must be 34 because all entries sum to 136. Verify rows, columns and diagonals before accepting a candidate.
Open Ganita Prakash Part I on NCERT · NCERT CIET Grade 7 resources