Careless smokers

Question

Cosmo complains: "At the party yesterday at our place, some guys were smoking in our living room. This morning we detected that these careless smokers had burnt four holes into the carpet."

Fredo: "What a shame!"

Cosmo continues: "The carpet is a square of side length 2.75 meters. The cigaret holes are tiny, essentially point-shaped, but clearly visible. We shall put the carpet to the garbage."

Fredo says: "But you could also cut the carpet down to a smaller square of side length 1 meters."

Cosmo: "I am not sure that this is possible. The cigaret holes are badly distributed."

Fredo: "It definitely is possible to cut the carpet down to a square of side length 1 meters."

Question: Is Fredo right?

Answer 1

Yes, it's possible, because one can fit 5 disjoint 1x1 squares in a 2.75x2.75 square: four in the corners, and one in the center rotated 45 degrees. The four cigaret holes can't eliminate all 5 squares.

The tilted square fits in a cross whose center's square diagonal has a of length 1. So the cross needs width $1/ \sqrt{2} \approx 0.707 < 0.75$.

So, the smallest square table with this property has sides length $2 + 1/ \sqrt{2}$. A smaller table might not work because putting cigaret burns just inside the four inside corners of the corner 1x1 squares eliminates all 1x1 subsquares.