Logical maths
You are given cards numbered 100, 99, 98, ..., 3, 2, 1 in that order. You want to rearrange them in the reverse order 1, 2, 3, ..., 98, 99,100 switching only two adjacent cards at a time.
What is the minimum number of switches necessary?
The answer is 4950.
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1 solution
card number 1 0 0 is switched between 9 9 cards before it, similarly card number n is switched between ( n − 1 ) cards.
Total number of switches is therefore ( 9 9 ∗ 1 0 0 ) / 2 = 4 9 5 0 , which is the sum of all individual switches.