Number Play · Lesson 4 of 10
Some Explorations in Grids
“Use row and column totals to solve number grids and spot impossible clues.”
• Interpret the outside numbers as sums of grid rows and columns. • Fill a 3 × 3 grid using 1–9 exactly once while meeting given totals. • Use minimum, maximum, and overall totals to reject impossible clues. • Explain why all row totals and all column totals each add to 45.
What do the outside circles mean?
Place the numbers 1 through 9 in a 3 × 3 grid, with each number used exactly once. A number written beside a row gives that row’s sum; one below a column gives that column’s sum. The clue does not tell us the position of every entry, so we combine it with the clues crossing the same cells.
| Left cell | Middle cell | Right cell | Row clue |
|---|---|---|---|
| 4 | 7 | 5 | 16 |
| 6 | 1 | 2 | 9 |
| 3 | 9 | 8 | 20 |
| Column total 13 | Column total 17 | Column total 15 |
The top row above has 4 + 7 + 5 = 16. The first column has 4 + 6 + 3 = 13. In each direction we count every cell once. Check the remaining row and column totals to see how the same nine entries produce two sets of clues.
Problem
The top row contains 4, 7, and a blank and totals 16. The blank’s column contains that blank, 2, and 8 and totals 15. What fills the blank?
- 1.The row requires 4 + 7 + blank = 16, so the blank is 16 − 11 = 5.
- 2.The column check gives 5 + 2 + 8 = 15, agreeing with the clue.
- 3.A candidate entry must satisfy every row and column clue that crosses it.
When is a grid impossible?
Before trying many arrangements, ask what a row of three distinct entries from 1–9 can total. The smallest possible sum is 1 + 2 + 3 = 6; the largest is 9 + 8 + 7 = 24. A clue of 5 or 26 is impossible immediately. A clue inside the range is only potentially possible; the other clues may still contradict it.
Problem
Could a column of three different entries from 1–9 sum to 26?
- 1.To make a column as large as possible, choose the three greatest available values: 9, 8, and 7.
- 2.Their sum is 24. No three distinct entries from 1–9 can total 26.
A total that never changes
No matter how the nine numbers are rearranged, adding the three row clues counts every number exactly once. Therefore the row clues add to 1 + 2 + ··· + 9 = 45. Adding the three column clues counts those same nine entries exactly once, so their total is also 45. If all six outside clues are displayed, their combined total is 90 because every cell is counted once by a row and once by a column.
Problem
Two row clues are 13 and 17. What must the third row clue be?
- 1.All row clues together total 45.
- 2.The missing clue is 45 − 13 − 17 = 15.
- 3.This total is within the allowable 6–24 range, although completing the grid still needs further checks.
Row clues adding to 45 is required, but it does not prove a particular set of row and column clues can all be satisfied with distinct entries. Check crossings and the no-repetition rule as well.
Quiz
What does a circled number at the end of a row represent?
What is the smallest possible sum of three distinct numbers from 1–9?
What must the three row clues together total?
What do all six row and column clues together total?
Row clues are 12, 16, and an unknown number. What is the unknown?
Practice Problems
- Verify all three row and column clues for the grid 4 7 5 / 6 1 2 / 3 9 8.
- Fill the blank in a row 8, blank, 3 whose row clue is 18. Check that the answer lies from 1–9.
- Explain why a row clue of 25 is impossible when the entries are distinct numbers from 1–9.
- If two column clues are 14 and 19, find the third. State whether the result is within the possible range.
- Draw any 3 × 3 arrangement of 1–9 and make its six row and column clues for a classmate.
- Can row clues 10, 14, 20 come from such a grid? Explain before trying to fill cells.
Key Takeaways
• Each outside clue is a row or column sum. • Three distinct entries from 1–9 have a sum between 6 and 24 inclusive. • All row sums total 45; all column sums also total 45. • The six outside sums total 90, but matching that total alone does not solve the grid.