Frodo discovered ancient writing on the walls of the Caverns of Euclidor. After careful examination, he discovered that each line consisted of a sequence of exactly 99 different runes. Each line was almost identical to the one immediately above it, with exactly one of the 99 runes changed, and the other 98 remaining the same. In total, 101 different runes were used in the entire text.

Not far from there, he found a door sealed shut. There was a line of 99 runes carved at the top and another one at the bottom of the door, but the lines in between were missing. Frodo understood he would have to carve new lines in between the two according to the rules observed earlier, in order to break the magic that seals it. He decides to do it using the fewest possible number of lines as he does not want to carve more symbols than he needs to. What is the maximum number of lines that he might need to add?

Mathsbombe Competition 2023 is organised by the The Department of Mathematics at The University of Manchester.
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