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?

  • 1
    $\begingroup$ Do you mean "Is it always possible to cut down a 1 meter square"? $\endgroup$ – Jet Sep 19 '15 at 19:26

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.

enter image description here

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.