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100 penguins (101 to 200); 21 penguins into holes (this approach: trying to think of if you have 21 penguins, you must have 4 that satisfy some property?)
--> 20 holes (promises 2 in one)
--> 10 holes (promises 3 in one)
--> 6 holes (promises 4 in "one")
--> 5 holes (promises 5 in one)
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different approach:
GIVEN 21 penguins, 21C2 = 210 different pairs; need to show that 2 pairs have same average r (i.e. sum 2r) (but two pairs could share 1 penguin so ehh -> wait jk, not possible for a+b and a+c to have same sum...)
------> 21 makes sense with my logic below because 20C2 = 190 which is less than 197 different sums
Values from 101 to 200 --> sum ranges from 203 to 399 (197 different sums) [side note: 100C2 = 4950 total sums]
197 different sum values to work with --> picking 21 penguins is like picking 210 different pairs --> by pigeonhole principle, 2 of these pairs must have the same sum --> side proof about how same sum = same average!
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