Input explanation

Problems and Contests
#1

I have solved Promotion Counting and want to ask for a simple explanation.

So, I made my own test case,

1 2
1 0
1 3
1 4.

So, obviously, there are 3 competitors who promoted to plat, and 2 + 3 = 5 promotions to gold, but for bronze to silver, we have 2 + 3 - 1 = 4. I want to know where the -1 comes from. We don’t necessarily overcount or anything. Thanks!

#2

Don’t know where you got 2+3-1 from. Isn’t it 3+4-1-1-1?

#3

I was thinking, 2 because there are an increase of two in the Gold, and 3 because there is an increase of 3 in Plat. With that same logic, there is a difference of -1 in Silver, so 2 + 3 -1. I just want to know why -1. BTW this logic passes 100%