A table of numbers has 4 columns and 99 rows and is created in the following way: starting from the second row, the quartet of numbers on each row is determined by the numbers in the previous row, specifically as the sum of the first and second numbers, the difference of the first and second numbers, the sum of the third and fourth numbers, and the difference of the third and fourth numbers.

a) What is the sum of the numbers on the 3rd row, if the sum of the numbers on the first row is 0?

b) What is the sum of the numbers on the 25th row, if the sum of the numbers on the first row is 1?

(K. Pazourek)
The entire table is determined by the numbers in the first row. The first few rows of the table look like this:

| 1 | $a$ | $b$ | $c$ | $d$ |
| :---: | :---: | :---: | :---: | :---: |
| 2 | $a+b$ | $a-b$ | $c+d$ | $c-d$ |
| 3 | $2 a$ | $2 b$ | $2 c$ | $2 d$ |
| 4 | $2(a+b)$ | $2(a-b)$ | $2(c+d)$ | $2(c-d)$ |
| 5 | $4 a$ | $4 b$ | $4 c$ | $4 d$ |
| 6 | $4(a+b)$ | $4(a-b)$ | $4(c+d)$ | $4(c-d)$ |
| 7 | $8 a$ | $8 b$ | $8 c$ | $8 d$ |
| 8 | $8(a+b)$ | $8(a-b)$ | $8(c+d)$ | $8(c-d)$ |
| 9 | $16 a$ | $16 b$ | $16 c$ | $16 d$ |
| $\vdots$ |  |  |  |  |

a) The sum of the numbers in the third row is twice the sum of the numbers in the first row. If the sum of the first row is 0, then the sum of the third row is also 0.

b) The sums of the numbers in the subsequent odd rows double progressively: the sum of the 5th row is 4 times the sum of the first row $\left(2 \cdot 2=4=2^{2}\right)$, the sum of the 7th row is 8 times the sum of the first row $\left(2 \cdot 4=8=2^{3}\right)$, the sum of the 9th row is 16 times the sum of the first row $\left(2 \cdot 8=16=2^{4}\right)$, etc.

Positive odd numbers are of the form $2 i+1$, where $i=0,1,2,3,4, \ldots$ The previous observations are generalized as follows: the sum of the $(2 i+1)$-th row is $2^{i}$ times the sum of the first row.

Since $25=2 \cdot 12+1$, the sum of the 25th row is equal to $2^{12}$ times the sum of the first row. If the sum of the first row is 1, then the sum of the 25th row is

$$
2^{12}=8^{4}=64^{2}=4096
$$

Evaluation. 1 point for preparatory observations (a few rows of the table); 1 point for the answer to question a); 2 points for the answer to question b); 2 points for the quality of the commentary.

Notes. Any of the expressions in (*), or any other equivalent expression, should be considered a correct answer to question b).

The sums of the even rows also double progressively. These sums are multiples of $a+c$, which is the sum of two numbers from the first row.