1.
Numbers 1, 2, 3, 4, etc., alternated with their square 1
1, 1*11=0, ... , 4, 4*41=15
2.
Alternate between dividing the previous number by two and multiplying the previous by three.
16, 16/2=8, 8*3=24, ..., 18*3=54
3.
1, 2, 3, 4, etc., after each two numbers insert their sum:
of each set is one greater than the first number of the previous set.
1, 2, 1+2=3, 3, 4, 3+4=7, ..., 5+6=11
4.
Start with 0. Add 1 to previous number, then 2 to previous, then 3, etc.
0, 0+1=1, 1+2=3, 3+3=6, 6+410, 10+515, 15+621, 21+728
5.
Alternate between dividing previous number by 2, dividing it by 4, and multiplying it by 6.
64, 64/2=32, 32/4=8, 8*6=48, 48/2=24,
24/4=6, 6*6=36, 36/2=18
6.
Each number is the sum of the previous two; if sum is > 100, then subtract 100.
35, 18, 35+18=53, 18+53=71, 71+53100=24, 71+24=95,
24+95100=19, 95+19100=14
7.
Change previous number by 1^{2}, 2^{2}, 3^{2}, 4^{2}, ...
The "change" alternates between adding and subtracting.
1,
1+1^{2}=2,
22^{2}=2,
2+3^{2}=7,
74^{2}=9,
9+5^{2}=16,
166^{2}=20
20+7^{2}=29
8.
Take prime numbers in descending order, starting at 23. The sequence is these prime numbers squared.
23^{2}=529,
19^{2}=371,
17^{2}=289,
13^{2}=169,
11^{2}=121,
7^{2}=49,
5^{2}=25
9.
Take the sequence 1, 2, 3, 4, 5, etc. and alternate between squaring and cubing the number; take the number modulo 100.
"x modulo y" means to take the remainder when x is divided by y. For the puzzles, we usually use modulo 100  this means to drop all but the tens and ones digits: 625 modulo 100 = 25.
1^{2}=1,
2^{3}=8,
3^{2}=9,
4^{3}=64,
5^{2}=25,
6^{3} modulo 100=16,
7^{2} modulo 100=49,
8^{3} modulo 100=12,
9^{2}=81,
10^{3} modulo 100=0
10.
Each number is calculated from the previous two as the absolute difference between them (that is, the larger number minus the smaller).
63,
74,
7463=11,
7411=63,
6311=52,
6352=11,
5211=41,
4111=30
