Because there are three kinds of socks, at least four socks must be pulled out to ensure a pair of socks. To ensure two pairs, you need to add two more pairs of socks, because if the first four pairs are only white, if the fifth pair is also white, there will be no two pairs of socks. That is to say, five socks have three colors, so touching a sock can definitely make a pair, and so on. The general formula is to get at least N pairs of socks.

3, segmentation, when n

F(n)= 1*n, 20<n & lt30, f(n)=2*n, .........., so the final sum =1+(1+...+9)+(1+)

4, 450 can't, 225 can, 225 =18+19+20+21+22+23+24+25+26+27. It is proved that a number can be expressed as the sum of 10 consecutive numbers. Y = a1+a 10 = ... = a5+a6, so x=5y, so x must be divisible by 5, and because a5 and a6 are adjacent, their sum must be odd, so one can be divisible by 5. As a result, odd numbers can be expressed as10 consecutive integers. =55, because natural numbers are needed, a1>; = 1。

5. The total volume is 4 3, = 64, and the volume of the cylinder without each surface is pi *12 *1= pi, so the remaining volume is V=64-6*pi=45. 16, and the original surface area is 6 * 4 2 = 96. Lateral area is pi*2* 1* 1=2pi, so the remaining surface area is s = 96+6 * 2pi = 96+12pi =133.7.

6, can't see clearly, please re-map.