My daughter wants to know, is there a term for even numbers which divide evenly into even numbers (that is, every other even number starting with zero). Is there anything else special about these numbers I haven’t noticed, other than it’s every other even number?
I’d call them “multiples of four” or “numbers with four as a factor”.
A mathematician would most often say that they are “divisible by four” or “congruent to zero modulo four.” The terms “evenly even” or “doubly even” fit the bill quite nicely, but neither of them are particularly common, I’d say, in the mathematical world.
Ancient numerologists used to call them “doubly even” but now we would just call them “multiples of four.”
I’ll go with “multiples of four” but “congruent to zero mod four” is also correct.
Speaking as a locksmith, I know one application of this. There’s a brand of combination padlocks which numbers their dials from 1 to 40 but they never use odd numbers in the combination. The weird part is that when you look at the combinations which are in use and ask yourself if each number is a multiple of four or not, the answers always go Yes-No-Yes or No-Yes-No. For example, if the third number in the combination is a multiple of four, then the first number will be also, but the second number won’t be.
…So, instead of having 64,000 combinations like it looks like they would, they only have 2,000? That’s pretty weak.
Wouldn’t it be less than two thousand? The fact that it’s in either a Yes-No-Yes or No-Yes-No pattern would eliminate some possibilities. 4-8-10, for example, would not be a possible combination.
I took that into account. 20 possibilities for the first number (which can be 0 or 2 mod 4), and then 10 possibilities each for the second and third (whose values mod 4 are constrained by the choice of first number).
I know nothing about what kind of lock you refer to, of course, but I remember back in high school, locker combination locks were often pretty flexible with numbers - if the combination specified 10, for instance, you could open it even if you dialed in 9 or 11. Could that be why they only use evens? If 10 and 11 are effectively the same, might as well just pick 10 by convention.
You are correct, they have 2,000 actual combinations even though the dial gives the impression that there ought to be 64,000.
Euler called them evenly even in his paper on Latin squares. Or at least his translator into French did. They were originally in Latin.
When I was in HS, I had two locks, one for my coat locker and one for my gym locker. One used the combination 7-43-36 and the other 9-41-36. I soon discovered that 8-42-36 would open both.
When I was in high school, the school had bought locks in batches of 40, with the dial in different places but otherwise the same. That is to say, if you had 7-24-17, then someone else had 8-25-18, and 9-26-19, and so on. Combine this with the fact that it was a small school, and the fact that the dial always turned the same amount when you closed the lock (after having been on the last digit to open it, of course), you had a fairly good chance of guessing a lock’s combination just by looking at the number the dial was currently on.