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The tile looks like pretty common indian slate. The issue was probably a void underneath, not properly back buttered before laying. The stone always comes in different thicknesses etc, so it's important to have 100% coverage to stop that happening.
Best option would be to grab a couple of spares from a tile shop and pop those in after a thorough clean/strip of the rest of the floor's residue and previous sealer.
Failing that, clean the loose out and fill with flexible grout and tone down with a sealer before sealing the rest of the floor.....it's a compromise though.
Looks like the previous owners dropped something heavy and smashed a hole (which has been gradually disintegrating) in these seemingly solid stone tiles. What’s the best option to repair/fill? I have a load of cement...
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