Concatenation (extra footage) - Numberphile

Two related videos...
Featuring Matt Parker...

Inder J. Taneja papers...
Single Digit Representations of Natural Numbers
Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9

Videos by Brady Haran

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NUMBERPHILE

[johnlou]

Concatenation:Ed Sheeran's next album.

[AaronHutson]

"you know I get really upset if I'm limited to base ten." lol

[PopeLando]

The only independent mathematical proof I ever produced was to show that a||b - (a+b)=0 (mod 9), AND that a||b - (a+b)[base B]=0 (mod B-1). It was to help a friend explain how a website was freaking him out by using a mathematical trick to "read his mind". A proud moment.

[MrDowntownjbrown]

To simply things, you could have a function that just concatenates a single digit, and use that repeatedly. The function would be a || b = 10*a + b. So to concatenate 17 and 29, you could do (17 || 2) || 9. For the solution to 10, 958, only single digit concatenation was required.

[AquilaSornoAranion]

So, allowing concatenation implicitly allows the use of the floor function. Then Matt could have just used the floor function on his first (not quite solved) solution. =)

[OwlOfBlues]

But is it okay to make a.b? Like 3 and 14 become 3.14? It's not exactly concatenation but feels very similar

[Reddles]

I've always thought of this kind of thing as more of a language puzzle than a real math problem. Sure, concatenation is a pretty strange and silly operation mathematically, but the entire problem only exists in the first place because of the specific way that we write our numbers. If we used a different base then we would have a different set of digits to use, and if we used a different system of writing numbers then the concept of digits might not make any sense at all.

Actually, i'm curious if there are similar puzzles for other writing systems. Maybe you could try to only use one of each roman number character or something...

[guitarraccoon]

I love how when you progressed through writing the operations, you felt a little more negative about it.

[Brindlebrother]

Hold up, I need to verify that 2 concatenated with 9 is in fact 29.


Yea he's right.

[NNOTM]

But, since in the original, it's only ever used to concatenate single digits, we could say that the equation is actually "10*a+b", which is a somewhat simpler equation. If we extend that to something like 17||29, we get 199, which clearly isn't concatenation anymore, but it _is_ consistent with the paper. And maybe there's a solution for 10958 with that as well.

[codebeard]

I think it's better to use ceil(log_B (b+1)) than floor(log_B (b))+1. They give the same value in all cases except b=0. The latter is not defined at b=0, while the former behaves consistently by preserving the number a in this case.

[SergeofBIBEK]

the formula is:
Number(a.toString() + b.toString());

Simple straightforward formula. It should totally be allowed.

[michael-gary-scott]

"You know I get really upset when I'm limited to base 10"

[BlankBrain]

We use concatenation when we perform long division. This is a trivial statement, but it supports why it is implicit.

[menachemsalomon]

I am wondering: Matt is using floor() + 1. Could he not more simply use ceiling()?

[DamianShaw]

Hi Matt, did you prove that you can't get 10958 without concatenation (either add, subtract, multiple &, divide; or with power)?

I happen to have already have a Python program which tackles this kind of problem and am exhaustively checking all possibilities for the 4 classic operators now. But looks like it will take the night to finish.

[badlydrawnturtle]

3 || 7 = 37
If equation is in base 10, the result translated to base 10 is 37.
If equation is in base 9, the result translated to base 10 is 34.
If equation is in base 58, the result translated to base 10 is 181.

Base-specific operations make me cry.

[ikarienator]

Except when the second number is 0. You can't get the log of zero.

[power-max]

I implemented a algorithm to do just this in javascript as part of a (failed) parser for a calculator applet project!

What I did was take 2 number integer inputs, multiply the first argument by the log of it and add it to the second one.

[wetlazer]

1:26 "Or whatever the base is..." because it's all about that bass, about that base.