First Idea, make a recursive loop, where if the value that's being looped squared is equal to the value from the list, it stops. A primative solution, but one that would work, except it wouldn't. The numbers given are far too large for that too work, and the loop reaches it's limit before getting the fourth number.
My second idea was to add a large number like 1,000 after the loop passes 5. This way, I can get farther before reaching the limit. I didn't even attempt this, the gap between the numbers is far too big to work.
I thought about using binary search, as the lesson was about this, but binary search is only used on sorted lists, here I'm working with a singe number.
My first loop didn't work because a the end of the code, I had it run the function again, resulting in the maximum recursion depth being reached. I'm gonna use a while loop instead so it's not stupid.
It wasn't stupid

def sqrt(N):
    i = 0
    while i < N:
        v = i * i
        if v == N:
            break
        i += 1  # Make an extra variable that is i times itself
    print(i)
    return i
from math import sqrt as sq
test_cases = [0,1,4,85248289,22297284,18939904,91107025,69122596,9721924,37810201,1893294144,8722812816,644398225]
answers = [int(sq(x)) for x in test_cases]

def checkValid():
    for i in range(len(test_cases)):
        if sqrt(test_cases[i]) == answers[i]:
            print("Check number {} passed".format(i+1))
        else:
            print("Check number {} failed".format(i+1))

checkValid()
0
Check number 1 passed
1
Check number 2 passed
2
Check number 3 passed
9233
Check number 4 passed
4722
Check number 5 passed
4352
Check number 6 passed
9545
Check number 7 passed
8314
Check number 8 passed
3118
Check number 9 passed
6149
Check number 10 passed
43512
Check number 11 passed
93396
Check number 12 passed
25385
Check number 13 passed
def collatzConjecture(N):
    print(N)
    while N != 1:
        if N % 2 == 0:
            N *= 0.5
            print(N)
        else:
            N *= 3
            N += 1
            print(N)
collatzConjecture(20)
20
10.0
5.0
16.0
8.0
4.0
2.0
1.0