Tricky Test 2 Level 71

Overview

• Difficulty: Hard
• Estimated moves: 1

The screen shows a 4-by-4 grid of numbers with one cell greyed out and marked "w". The grid reads:

```
1 1 1 1
1 3 5 7
1 5 w 25
1 7 25 63
```

Below the grid is an input with minus/plus buttons showing a number, and an "OK" button. The prompt is simply "Find w." This is a number-pattern puzzle; the grey cell is the unknown value you must deduce and type in.

Step-by-Step Solution

1. Notice that the matrix is symmetric across its main diagonal: f(1,2)=5 matches f(2,1)=5, f(1,3)=7 matches f(3,1)=7, and f(2,3)=25 matches f(3,2)=25.
2. Identify the recurrence rule: each cell equals the sum of the cell above it, the cell to its left, and the cell diagonally up-left. In formula terms, f(i,j) = f(i-1,j) + f(i,j-1) + f(i-1,j-1).
3. Verify the rule on known cells: f(1,1)=1+1+1=3; f(1,2)=1+3+1=5; f(2,3)=5+7+5=17? Check instead: f(2,3)=f(1,3)+f(2,2)+f(1,2). Working out w first:
4. Compute w = f(2,2) = f(1,2) + f(2,1) + f(1,1) = 5 + 5 + 3 = 13.
5. Confirm: f(2,3) = f(1,3) + f(2,2) + f(1,2) = 7 + 13 + 5 = 25, which matches the grid. f(3,3) = f(2,3)+f(3,2)+f(2,2) = 25+25+13 = 63, which also matches.
6. Use the plus/minus buttons to set the input to 13, then press OK.

Strategy & Reasoning

The grid is a table of Delannoy numbers, which count paths on a grid and follow exactly the recurrence f(i,j)=f(i-1,j)+f(i,j-1)+f(i-1,j-1) with boundary values of 1. The boundary row and column are all 1s, which matches the outside edges of the given grid. Once you spot that each interior cell is the sum of the three neighbors above, left, and diagonally up-left, the unknown cell is unambiguous. The value 13 is the only number that makes the rest of the grid consistent; any other guess breaks one of the surrounding cells.

The input at the bottom has minus and plus buttons flanking a number display, with an OK button. You press plus repeatedly to raise the value to 13. The symmetry of the matrix is a useful check: because f(i,j)=f(j,i), the cell at row 3 column 2 equals the cell at row 2 column 3, both shown as 25. This symmetry confirms you are using the correct recurrence rather than a row-by-row arithmetic rule. Working from the top-left corner outward, the center cell naturally lands on 13, and the bottom-right 63 validates the whole table.

Tips & Common Mistakes

• Looking for a simple row/column arithmetic sequence. Rows do not follow a constant difference or ratio; you need the three-neighbor sum rule.
• Forgetting the diagonal term. The recurrence adds the up-left cell as well, not just up and left.
• Miscounting indices. The top row and left column are all 1s; start the recurrence from those boundaries.
• Typing the wrong value. The answer is 13, not 11, 15, or 17.