B3. We have a square sheet of paper on the table that is divided into $8 \times 8=64$ equal squares. These squares are numbered from a1 to h8, similar to a chessboard (see fig. 1). We are now going to fold, with the square a1 always remaining in a fixed position on the table. First, we fold along the horizontal middle (fig. 1). As a result, square a8 ends up on top of square a1. Next, we fold along the vertical middle (fig. 2). Then we fold again along the newly formed horizontal middle (fig. 3), and so on. After six folds, we have a packet in front of us (fig. 7) that can be considered a stack of 64 square pieces of paper.

![](https://cdn.mathpix.com/cropped/2024_04_17_6ebf2d3f59be4ec6e677g-1.jpg?height=365&width=389&top_left_y=2096&top_left_x=248)

fig. 1

![](https://cdn.mathpix.com/cropped/2024_04_17_6ebf2d3f59be4ec6e677g-1.jpg?height=257&width=362&top_left_y=2207&top_left_x=750)

fig. 2

![](https://cdn.mathpix.com/cropped/2024_04_17_6ebf2d3f59be4ec6e677g-1.jpg?height=214&width=255&top_left_y=2246&top_left_x=1226)

fig. 3

![](https://cdn.mathpix.com/cropped/2024_04_17_6ebf2d3f59be4ec6e677g-1.jpg?height=86&width=86&top_left_y=2373&top_left_x=1739)

fig. 7

We number the squares of the resulting stack from bottom to top, from 1 to 64. Square a1 thus gets number 1.

Which number does square f6 get?
B3. 43 In the picture below, the positions are indicated where square f6 ends up when folded. We start at f6 (after 0 folds). Then our square ends up at positions f3, c3, c2, b2, b1, and a1.

Each time we fold, the order of the stack is reversed: the squares that were below f6 end up above f6, and vice versa. Moreover, the stack is placed on top of an equally high stack of other squares. After $k$ folds, the stack consists of $2^{k}$ squares. If there are then $a$ squares above f6 and $b$ squares below f6, then after $k+1$ folds, there will be $b$ squares above f6 and $a+2^{k}$ squares below f6. This gives us the following table.

| Number of folds | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| Squares in stack | 1 | 2 | 4 | 8 | 16 | 32 | 64 |
| Squares above f6 | 0 | 0 | 1 | 2 | 5 | 10 | 21 |
| Squares below f6 | 0 | $0+1=1$ | $0+2=2$ | $1+4=5$ | $2+8=10$ | $5+16=21$ | $10+32=42$ |

After six folds, there are 42 squares below f6, so f6 gets number 43.