Don't just memorize all 1000 cards! There's a trick to learn!
First, memorize 0^3 through 10^3 (tagged CubeTrick1). Notice that the ones place matches for 0, 1, 4, 5, 6, and 9, and swaps between 2/8 and 3/7.
Second, to find the root of any cube up to 1000000:
1. The tens place is given by ignoring the last three digits; of what remains, take the largest root without going over.
2. The ones place is given by the "notice" above.
Some examples:
cube root of 226981
1. Ignore "981", leaving 226. 6^3 is below; 7^3 is too big.
2. 1 remains the same.
answer is 61
cube root of 804357
1. Ignore "357", leaving 804. 9^3 is below; 10^3 is too big.
2. 7 swaps with 3.
answer is 93
cube root of 5832
1. Ignore "832", leaving 5. 1^3 is below; 2^3 is too big.
2. 2 swaps with 8.
answer is 18
Practice with cards tagged CubeTrick2.
Now, ask a friend to pick any number from 10 to 100, cube it, and tell you the result. You can instantly tell them the original number.