Solution to Challenge 1

Folding and measuring
1. Fold the square corner to corner, crease and unfold, to mark the diagonal.
2. Fold the two edges that meet at one corner in so they lie along the diagonal. Crease firmly.
3. Unfold the paper. Each crease runs from that corner to the opposite edge. Measure along that edge from the corner whose edge folded in to where the crease meets it.
4. The expected distance is the side length times tan(22.5 degrees), about 0.414 times the side. For a 150 mm square that is 62.1 mm, and for a 210 mm square 87.0 mm. Within a millimetre or two is good folding.

Changing the script

import math

for s in (100, 200):
    d = s * math.tan(math.radians(22.5))
    print('Square', s, 'mm -> crease meets the far edge', round(d, 1), 'mm from the corner')

Output when run:

Square 100 mm -> crease meets the far edge 41.4 mm from the corner
Square 200 mm -> crease meets the far edge 82.8 mm from the corner

What happens when the paper is twice as wide
The distance doubles too (41.4 to 82.8). The crease distance is always the same fraction, about 0.414, of the side, so every length in the fold scales with the paper. Bigger paper is easier to fold accurately, and the shape is identical.

Checkpoint: the two creases are mirror images of each other, and the corner angle between them is 45 degrees.