The Game of Differences - Colegiul Național din Iași (Iași - Roumanie)

Research topic:
There are 4 natural numbers a, b, c, d written on a row. The differences |a-b|, |b-c|, |c-d|, |d-a| are then written on the next row. The process is continued:

7 3 9 2
4 6 7 5
2 1 2 1
1 1 1 1
0 0 0 0

It can be observed that the null row 0, 0, 0, 0 has been obtained.
 Is this a coincidence or is the null row obtained for any natural numbers a, b, c, d?
 What happens if the rows don’t have 4 numbers, but 3, 5, 6,...?
 What happens if we have 4 rational numbers
 What happens if we have 4 real numbers

Brief presentation:

• We solved the natural numbers’ case using remainders modulo 2.
• We also treated the cases of 3, 4, 5, 6, 7, 8, … natural numbers on the first row, obtaining different conclusions, afterwards we made a C++ program.
• We extended the case of integers, followed by rational numbers.
• Subsequently, we prove significant results for real numbers.