Would you elaborate on that? That doesn't make any sense to me

Multiplication is repeated addition.

2x3= 2+2+2 AxB= A+A B times.

Raising to a power is repeated multiplication.

2^3= 2x2x2 A^B = AxA B times

Roots are the inverse of powers

Division is the inverse of multiplications.

Given this… the OC’s comment about giving exceptions to powers in the rules is nonsense because they are still operations.

Multiplication is not the same as repeated addition, which you can observe as soon as you try the two different operations on fractions or negative numbers. For example, you can't add a number to itself 1/3 times, but you can certainly scale a number by 1/3.

Multiplication is a scaling operation.

Can you not? 9+(9(1/3)) [adding nine to itself a third of a time] is nine plus one third of nine {or nine divided by three} which is one of the three in three times three that equals nine which is three therefore 9+(9(1/3))=11 because 9+3=11

9+3=12, but otherwise I agree

:)

I like it when people who are clearly smart do things like this! Makes me feel better.

Reminds me of this story: https://news.ycombinator.com/item?id=5663709

That's Grothendieck multiplication.

By dividing by 3 you are introducing a scaler into the operation.

Multiplication also flips signs (direction). That's why a negative * negative = positive, whereas a negative + negative = negative. The addition operator does not flip direction.

Multiplication is not generally the same as multiple addition operations. However, it has the same result for positive whole numbers.

Multiplication is the generalization of repeated addition. Repeated addition can be generalized to fractional values.

In the integers, it is the same thing, with a rule for negative times.

In the rational numbers, you can generalize to adding a number to itself a rational number of times. Adding a/b to itself p/q times is defined as adding a to itself p times, with a divisor that is adding b to itself q times.

So adding the rational 5 (5/1) with itself 2_and-1/3 (4/3) times just means (5+5+5+5) / (1 + 1 + 1) = 20/3, a rational that we can also write as 6 -and-2/3. Only addition of integers is used.

Which is the same as rational multiplication (a/b)(p/q) = (ap)/(b*q).

But it's true that you can't say that rational multiplication is the same as _integer_ repeated addition.

>you can't add a number to itself 1/3 times,

Learn logarithms.

Multiplication is adding numbers to themselves. Exponentiation is multiplying numbers by themselves. Ditto division/taking roots. Squaring a number is just the instance of exponentiation that happens to have two operands. If taking a square root is an inherent use of a two, then dividing a number by another number should also be a use of a two, because there are two operands.

Because it's just wrong. hahajk said that taking the 2nd root spends a 2, not that "roots are invalid".

I think you misunderstand. They’re saying (maybe not very well) that the square root is not really a unary operator, it’s a binary operator of which the “square root” symbol just elides the 2 in the left.

But the “rules” of this game are necessarily arbitrary.

Exponentiation is to multiplication as multiplication is to addition. Squaring is to multiplication as doubling is to addition.

> Squaring is to multiplication as doubling is to addition.

as halving is to addition.

Square rooting is to multiplication as halving is to addition.

GP didn't say roots are invalid, just that they burn a two.

If we agree that 2^2 uses two twos (and, indeed, that 2^3 isn't valid, because it uses a 3), then what's the equivalent of a square root?

Of course to my simple math brain, a square root is ^1/2, which includes the digit 1. But I agree there is someone 2-ish about the square root.

It can be written as ²√2 and there's also ³√2 and so on

Yep.

It would be like if we had a stylistic convention that you can drop the second number in an addition and it becomes 1. So "6 +" means "6 + 1."

Now I can trivially make any number, but I'm just 'exploiting' a convenience feature of the notation:

0 = 2 + 2 - 2 - 2

1 = 2 + 2 - 2 - 2 +

2 = 2 + 2 - 2 - 2 + +

Etc

Yes, taking the nth root is an operator. Taking the SQUARE root is this operator with the operand 2, which is usually graphically omited. I totally agree with the top comment: it's burning one of the 2's.

The obvious counterargument is that the problem statement is

> Given exactly four instances of the digit 2 and some target natural number, use any mathematical operations to generate the target number with these 2s, using no other digits.

and the square root symbol is a mathematical operation and is not a digit. That it is equivalent to taking the 2nd root isn't relevant ... the whole puzzle is about finding and using equivalents.