You’re right - I kind of stupidly overlooked that. It also has the interesting property that the resulting numbers all have the same digit order, in a sort of loop with each number slicing the loop in a different place. 012345679 includes all such rotations of itself as well, amonst its 81 multiples.
I hadn’t realized that 12345679 was so un-unique in this regard (which is a little disappointing, actually). I don’t suppose you can tell me the next largest number with this property?
These math ones make my head hurt. Here are some word ones, also car talk puzzlers. The first one is definitely lateral thinking; I don’t know if the second one would remotely count, but it’s cool, so there.
First: think of a word that’s a plural. Add an “s” to the end of it. Now you have a singular word. What are the two words?
princes–>princess. There may well be other answers that follow this basic pattern, but I can’t think of any.
Second, also a plural: English has a lot of irregular plurals for animals (goose–>geese, for example). Think of a plural for an animal that doesn’t share a single letter in common with the singular form.
kine–>cow
Well, it depends on what exactly the property you’re referring to is. If you mean digit-sequences which include all rotations of themselves among their multiples modulo an equal length string of 9s, I’m afraid every digit-sequence has this property; in an arbitrary base b, if k has d many digits, then bk mod d repetitions of the digit (b - 1) = the digits of k rotated one position “leftwards”. (It’s easy to see that all but the leftmost digit will move to the left; the reason the leftmost digit moves back to the unit position is because b[sup]d[/sup] = 1 + d repetitions of the digit (b - 1), and thus b[sup]d[/sup] mod d repetitions of the digit (b - 1) = 1).
But I imagine you’re probably thinking of a more distinctive property than that.
Er, just to stave off unlikely misparsings by readers, everywhere I wrote “d repetitions of the digit (b - 1)”, that phrase binds more tightly than any surrounding operators.
Which means that once you’ve memorized the decimal expansion for 1/7, you also know 2/7, 3/7, 4/7, 5/7, and 6/7. For instance, if someone asks you what 3/7 is, you can say to yourself “Well, that’s a little less than half, so it has to be the one that starts with 4.” Then you say “1” to yourself, and “point 42857” out loud, then start repeating.
Yes, I know that I have weird parlor tricks.
Just to generalize slightly from 7 in base 10, the analogous property holds of p in some base just in case p is prime; furthermore, the specific bases where it holds will be those which are primitive roots modulo p [i.e., those bases b such that p - 1 is the length of the shortest string of (b - 1)s which p divides].
So, for example, 3 has this property in base 2 (and any base equivalent to 2 modulo 3), as do 5 and 11 (in bases equivalent to 2 modulo 5 or 11, respectively), 7 has this property in base 3 (and thus in decimal, as 10 mod 7 = 3) as does 17, 23 has this property in base 5, 41 has this property in base 6, and so on.
Whether calculating the decimal expansion of n/11 in base 13 makes as good a parlor trick, I couldn’t say…
You know, if you keep making me feel stupid… :mad:
I suppose the property I’m looking for is a lack of multiples with non-notable digit arrangements. Obviously “non-notable” is not terribly well defined here, but if we look at, say, 4571:
mod 9999
1 4571 4571
2 9142 9142
3 13713 3714
4 18284 8285
5 22855 2857
6 27426 7428
7 31997 2000
8 36568 6571
9 41139 1143
10 45710 5714
We do see the rotation at the multiple of 10, but the other multiples are nothing to write home about (except maybe 2000). And as you note the rotation is more about mod (10[sup]n[/sup]-1) than anything 4571 is doing.
But I get that a question this poorly defined doesn’t deserve an answer, so I’ll just shut up about this now.
Ah, well, how about this: after 142857, the next numbers with the property that all of their multiples are either 0 or rotations of themself (modulo the same-length string of 9s) are 0588235294117647, 052631578947368421, 0434782608695652173913, 0344827586206896551724137931, and 0212765957446808510638297872340425531914893617, which have the same relations to 17, 19, 23, 29, and 47, respectively that 142857 has to 7.
That’ll do - thanks!
And you remain a math god.
(The next numbers that aren’t mere rotations of previous ones, that is…)
That reminds me of another nice one:
Write out all the integers up to ten billion, using the full names of the numbers, like “one”, “two”, …, “one hundred”, “one hundred one”, …, “nine hundred ninety-nine”, “one thousand” etc.
(We’ll use that convention for naming numbers, so it’s “one hundred one” rather than “a hundred and one” or anything like that. Also, no “twelve hundred” etc. - that would be “one thousand two hundred”.)
Sort the list of numbers alphabetically, ignoring spaces and punctuation. What is the first odd number in the list?
eight billion eight hundred eighteen million eight hundred eighteen thousand eight hundred eighty five?
Not that more examples are necessary, but inbetween those last two, there is a number with not quite the same property but an interesting one nonetheless: (10^(42) - 1)/49 = 020408163265306122448979591836734693877551, whose reduced multiples are all rotations of it, except when it is multiplied by a multiple of seven, in which case it produces seven repetitions of the corresponding multiple of 142857.
You have two boxes that are Norton equivalent and Thévenin equivalent circuits.
One has a constant current source of 1 Amp with 1 Ohm resistor across it. Two terminals outside the box connect across the resistor.
The other box has a constant voltage source (1 Volt) with a 1 Ohm resistor in series. Two terminals outside the box connect between the open end of the resistor and the negative terminal of the voltage source.
Electrically, the two circuits are equivalent.
Without opening the either of the two boxes, how can you tell which has the constant current source and which has the constant voltage source?
Touch them. The constant current source is generating 1W of power and will be warmer.
Which is probably why, in practice, electrical systems are usually designed to be based on constant-voltage sources, not constant current sources.
I like it–I got the “eight billion” part, was thinking “eight billion and one,” since I hadn’t read the post carefully. Indistinguishable’s answer looks right.
But it is not.
… eight, surely.
Though I think I would alphabetize “Eight billion” before that. And “eight” before “eight billion”.
Eight isn’t odd.
Ximenean, are you sure the answer is a, for lack of a better term, fair answer? Would you mind putting the answer in spoilers or else providing it outright?
You didn’t overlook the part about ignoring spaces and punctuation marks? We are sorting purely on the string of letters.
