Imagine two concentric roulette wheels, each divided into 100 sectors. Choose 50 sectors at random on each wheel, paint them black, and paint the rest white. Prove that we can now position the wheels so that at least 50 of the aligned sectors match.
Follow a sector on the inner wheel through a complete revolution: That sector will find exactly 50 matches. The same is true for each of the 100 sectors on the wheel; altogether, as the wheel turns through 100 positions, there will be 100 × 50 = 5,000 matches. This means that the average number of matches per position is 50, so a position must exist with at least 50 matches.