How does this algorithm make the character jump in pygame?

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At the started Jumpcount is set 10.

Jumpcount = 10

The jump runs until Jumpcount is less or equal -10. Therefore, a jump takes exactly 21 cycles:

if Jumpcount >= -10:

Neg is “signe” of Jumpcount. It is 1 if Jumpcount is greater or equal to zero and -1 otherwise:

Neg = 1
if Jumpcount < 0:
    Neg = -1

In each frame, the player’s y coordinate is changed by the quadratic function (Jumpcount ** 2) * 0.5.

y -= (Jumpcount ** 2) * 0.5 * Neg

Since this term is multiplied by Neg, it is positive if Jumpcount is greater than 0, and 0 if Jumpcount is 0, otherwise it is less than 0.
When the amount of Jumpcount is large, the change in the y coordinate is greater than when the amount is small. See the values for (Jumpcount ** 2) * 0.5 * Neg in the 21 cycles:

50.0, 40.5, 32.0, 24.5, 18.0, 12.5, 8.0, 4.5, 2.0, 0.5, 0.0, 
-0.5, -2.0, -4.5, -8.0, -12.5, -18.0, -24.5, -32.0, -40.5, -50.0

At the beginning the values are positive and the player jumps. In the end the values are negative and the player falls down.
The sum of this values is 0. Therefore the y-coordinate has the same value at the end as at the beginning.

See also How to make a character jump in Pygame?.

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