Question 46
Teacher Jacira proposes a game on the board:
I) A student writes a sequence of $n$ nonzero digits.
II) A second student writes a sequence of $n-1$ digits whose first entry is the positive difference of the first two digits in the first sequence, whose second entry is the difference of the second and third digits, and so on.
III) A third student uses the same process on the second sequence.
IV) Continue until the final student writes a single digit.
V) If a zero appears in any sequence, call it a FAILED SEQUENCE and stop.
For example:
| Student 1 | 5 | 8 | 9 | |
| :---: | :---: | :---: | :---: | :---: |
| Student 2 | 3 | | | 6 |
| Student 3 | | 2 | 5 | |
| Student 4 | | | | |
After several games, the teacher proposes these challenges:
a) Continue the sequences below, given the first student's sequence.
| Student 1 | 1 | 2 | 4 | 8 |
| :--- | :--- | :--- | :--- | :--- |
| Student 2 | | | | |
| Student 3 | | | | |
| Student 4 | | | | |
b) Construct an example of a game with 5 sequences ending in the digit 5.
c) Explain why there cannot be a game with 6 sequences ending in the digit 5.
I) A student writes a sequence of $n$ nonzero digits.
II) A second student writes a sequence of $n-1$ digits whose first entry is the positive difference of the first two digits in the first sequence, whose second entry is the difference of the second and third digits, and so on.
III) A third student uses the same process on the second sequence.
IV) Continue until the final student writes a single digit.
V) If a zero appears in any sequence, call it a FAILED SEQUENCE and stop.
For example:
| Student 1 | 5 | 8 | 9 | |
| :---: | :---: | :---: | :---: | :---: |
| Student 2 | 3 | | | 6 |
| Student 3 | | 2 | 5 | |
| Student 4 | | | | |
After several games, the teacher proposes these challenges:
a) Continue the sequences below, given the first student's sequence.
| Student 1 | 1 | 2 | 4 | 8 |
| :--- | :--- | :--- | :--- | :--- |
| Student 2 | | | | |
| Student 3 | | | | |
| Student 4 | | | | |
b) Construct an example of a game with 5 sequences ending in the digit 5.
c) Explain why there cannot be a game with 6 sequences ending in the digit 5.