## Order of magnitude puzzle: Android sign-ins

One of the security options with Android lets you sign in by dragging you finger to trace a pattern you’ve chosen on a 3×3 square numbered 1-9. The codes have to be at least 4 digits long, you can’t repeat any digit, and you can’t lift your finger off the pad. To my always-wrong intuition, that seems like it affords too few possibilities. So, your task is to guess (or, if you must, figure out) roughly how many choices there are.

[Semi-Spoilers] You start with with the following range of numbers: 1,234 to 987,654,321. That is a boatload of numbers. But you remove all the numbers that have repeated digits. For a 9-digit number, there are only 362,880 numbers (9 factorial) without repeated digits, so that’s like subtracting 100 million numbers from the mix. Our son Nathan says that it’s the same number for all the 8-digit possibilities, because 8 factorial x 9 is the same same as 9 factorial. (I’m lost. Ask him.) After you do all of them down through 4-digits, you have to subtract the sequences that have non-contiguous numbers (based on the 3×3 square). So, it’s a big number, especially since the Android UI puts in a time-out after 10 wrong tries. But it’s not an astronomical number. I’m guessing it’s under a million.

But I fully expect to be shown to be wildly wrong.