Starting with 2,0,0,3, we construct the sequence 2,0,0,3,5,8,6,..., where each new digit is the mod10 sum of the preceding four terms. Will the 4-tuple 0,4,0,7 ever occur?
Any help is greatly appreciated. Since this problem is from a Working Backwards chapter, I have tried working with the desired four tuple and working backwards to determine other previous integers with no contradiction apparent. Thus, I am greatly confused.
On
[Lemma] The sequence will come back to the original tuple (in this case, 2003).
(Proof)
Choosing the next number is always deterministic. At the same time, identifying the previous number is also deterministic. As there are finite set of 4 digit numbers, at some point the sequence will run out. And then it will have to cycle. The initial number should be inside that cycle, as otherwise it would mean that the cycle has a branch (or a junction) that is not possible as the backward sequence is also deterministic.
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So, 2003 will show up 2nd time (and actually over and over again). For 2003 to show up the below sequence should happen:
$$0-4-0-7-1-2-0-0-3$$
Therefore, 0407 will show up.
Okay, from any x, y, z, w we can determine the previous term and there is no choice for variation. (As A + x +y + z = w has only one answer.)
As there are only finitely many 4-tuples the sequence must eventually repeat.
2,0,0,3 must come from
1,2,0,0 which must come from
7,1,2,0 which must come from
0,7,1,2 which must come from
4,0,7,1 which must come from
0, 4, 0,7.
Thus the sequence must repeat and 0,4,0,7 will appear near the very end of the cycle.