Mouse Trap

Number twenty-one cards, 1, 2, 3, etc., up to 21, and place them in a circle in the particular order shown in the illustration. 
These cards represent mice. 
You start from any card, calling that card "one," and count, "one, two, three," etc., in a clockwise direction, and when your count agrees with the number on the card, you have made a "catch," and you remove the card. 
Then start at the next card, calling that "one," and try again to make another "catch." And so on. 
Supposing you start at 18, calling that card "one," your first "catch" will be 19. 
Remove 19 and your next "catch" is 10. 
Remove 10 and your next "catch" is 1. 
Remove the 1, and if you count up to 21 (you must never go beyond), you cannot make another "catch." 

Now, the ideal is to "catch" all the twenty-one mice, but this is not here possible, and if it were it would merely require twenty-one different trials, at the most, to succeed. 
You may make any two cards change places before he begins. 
Thus, you can change the 6 with the 2, or the 7 with the 11, or any other pair. 
This can be done in several ways so as to enable you to "catch" all the twenty-one mice, if you then start at the right place. 
You may never pass over a "catch"; you must always remove the card and start afresh.

See answer

Math Genius