I used to be single, dig it? Then I found someone else who was single, and now we're in a relationship. The thing about relationships is that it's more than the sum, or even the multiple, of its parts. We're just... together.
Now, we've got neighbors - a couple in a relationship like ours. You know what I mean by neighbors, right? We're very close. They're just one away.
So, one day we'd invited them over for dinner, and we were being a little silly with the champagne and toasting each other. So I decided to toast myself - to raise to myself. My partner did the same thing to herself, and both of them did too. And suddenly it was like we were all in a relationship together - like me and my partner were together with the two of them. And all because of the added effects of me raising my glass to myself, my partner raising her glass to herself, and so on.
Hey, are you zoning out? Oh boy, you weren't paying any attention, were you? Do you remember our names, at least?
Your names are
3, 4, 3, and 5
From the first line,
You used to be a single digit. Now you are in a relationship -- not additive or multiplicative, but side by side.
Your neighbors are
just one away, so either one greater or one less than you.
The final relationship
must have four digits, so we need a four digit number that results from taking the individual four digits from each two-digit number and taking them to their own power. If lesser couple is XY, then the other is XZ with Z = Y+1. To get a four digit number, X Y and Z must be less than 6 (because 6^6 = 46656). We also require a 5 because 4^4 = 256, so a 4-digit number would only result if everyone is 4 (but this is a contradiction since one person is Y and another is Y+1). Taking 5^5 = 3125, it is clear that one digit must be 3, ie the leading digit of 3125, and there can only be one 5 or else there will be a digit greater than 5. We now know two digits, 3 and 5, and another must be 4 because Z = Y+1 (with Y=3 and Z=4 or Y=4 and Z=5). Since there is only one 5, Z=5, Y=4, and X=3.
Everybody knows a couple where both members have the same name... and in my case, that's you guys; because you are both named:
ONE (1) - after all, you were single (i.e. 'one'), and so was your partner - another 'one'.
Now that you're in a relationship, together you are:
ELEVEN (11) - the two of you standing side-by-side, more than the sum (1+1=2<11) or even the multiple (1x1=1<11) of your parts.
Your neighbours from one door down ('just one away') are:
TEN (10=11-1) (I presume you live on a crescent or in a very old part of town...) i.e. another One (what are the chances?!) and a Zero.
When you toast (i.e. raise to yourself), one thing changes:
You, your partner and one of your neighbours are unchanged, since you are all '1 raised to the power of 1' (i.e. still 1). However, your other neighbour - Zero (0) - when raised to themselves becomes 0-to-the-power-of-0, which is 1.
Everybody is 1, and you are all one with each other, which rather sounds like you're all in a relationship together!