yes, the largest number of squares a larger square cannot be partitioned into is 5(this was a national mathcounts problem.) outline of a proof: if i can create a partition into n squares, an n+3 square partition is possible by cutting one of those small squares in 4. i can make a partition with 4 squares in any square, and thus 7 by the above. 8 is possible(in a 4 by 4 grid, make a 3 by 3 and let the rest be unit squares-this adds up to 1 3x3, 7 1x1 i.e. 8 squares.), 9 is possible(3 by 3 division), and thus every partition into a higher number of squares is possible. edit: suppose it were possible and there were a 1by 1 square in the partition. then there must be at least 3 other 1 by1's. this all together makees a 2 by 2, and then there must be at least 3 more 2 by 2's, and basically you get a sum 4+3+3+3...... try to work with that....