Earlier today I set you this puzzle about PINs, the four-digit passcodes we use for phones and bank accounts.
In the comments, many people said that the answer was obvious. These people fell into the trap.
Perplexing PINs
You and a friend are discussing how you choose four-digit PINs. You establish that neither of you would ever use the digit 0.
“I like to choose four different random digits,” you say.
“I like to choose three different random digits,” they reply, “so one of the digits is used twice.”
Which strategy gives the largest pool of possible four-digit PINs?
Answer: Surprisingly, both strategies create exactly the same number of possible PINs.
Show me your workings: Your strategy is to choose four different nonzero random digits. There are nine choices for the first digit, eight for the second digit, seven for the third and six for the final digit. The number of possible PINs is thus 9 x 8 x 7 x 6 = 3024.
Your friend’s strategy is to choose three different nonzero random digits, thus repeating one of them. There are 9 choices for the digit that is repeated. And there are 6 positions for the repeated digits:
-
xx–
-
x-x-
-
x–x
-
-xx-
-
-x-x
-
–xx
There are 8 choices for the leftmost other digit, and there are 7 choices left for the final digit. So the total number is again 9 x 8 x 7 x 6 = 3024.
By mathematical coincidence, the number of possibilities when you choose three or four number is exactly the same. Nice!
I hope you enjoyed this week’s puzzle. I’ll be back in two weeks.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
I’m the author of several books of maths and puzzles, and also the co-author with Ben Lyttleton of the children’s book series Football School. The latest in the Football School series is The Greatest Ever Quiz Book, out now!
I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.
