KA-0105Patterns in tables and grids
4 points, difficulty 2 of 3Level 3–4about 90sEach row of a grid starts 3 more than the row above. Each cell is 1 more than the one to its left. The top-left cell is 1. What is in row 4, column 3?
Text description of the figure
A grid four rows by four columns. The first row reads 1, 2, 3, 4 and the second row reads 4, 5, 6, 7. The remaining rows are blank, and the cell in row 4, column 3 is shaded.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Find where row 4 starts, then move across.
2The strategy
Use the row rule to walk down, then the column rule to walk across. Count the steps between rows and between columns, not the rows and columns themselves.
3The full solution
Row 1 starts at 1, and row 4 is three steps down: 1 + 3 x 3 = 10. Column 3 is two steps across: 10 + 2 = 12.
Solution
The reliable way
Walk down first, then across, counting steps rather than rows. From row 1 to row 4 is 3 steps of 3, so row 4 starts at 1 + 9 = 10. From column 1 to column 3 is 2 steps of 1, so the cell holds 10 + 2 = 12. Check the rule on a cell you can already see: row 2 column 1 should be 1 + 3 = 4, and it is. The transferable idea: test a rule on a cell you can see before trusting it on one you cannot.
The elegant way
The cell in row r, column c holds 3(r - 1) + c, so row 4 column 3 holds 9 + 3 = 12.
Why this is on the test: Two rules acting at once is where students count rows instead of steps, and checking against a visible cell is the habit that catches it.