i know everyone plays with their own rules, but from my perspective, a square root spends a 2 - it's not an operator like + or ! . there's an implicit "2" since you could also take the cube root or fourth root
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 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.
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
Because it's just wrong. hahajk said that taking the 2nd root spends a 2, not that "roots are invalid". (FWIW, I'm not necessarily endorsing the view that the square root symbol implicitly spends a 2, I'm just saying that the counterarguments given are logically wrong ... actually more like semantically wrong.)
The comment below by drfloyd51 is similarly wrong ... hahajk didn't say anything about "giving exceptions to powers in the rules".
And the comment by speglt is also wrong:
> 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.
The number of operands isn't relevant; the degree of the root is.
And from SideQuark we get
> One could even claim division is raising to the -1 power, thereby disqualifying it in exactly the same manner you disqualify sqrt.
But hahajk said nothing about anything being disqualified. (Not to mention that division is not raising to the -1 power ... raising what? n/d = n * d^-1, but this is completely irrelevant to hahajk's argument.)
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.
Alternatively, one can claim sqrt is a single symbol, and there are extensions requiring a digit. Log, ln, also have hidden bases, but they’re also valid symbols. Cos/sin may be in radians or degrees, both in common use, and no real place to denote it.
One could even claim division is raising to the -1 power, thereby disqualifying it in exactly the same manner you disqualify sqrt. I’ve never seen sqrt with a 2 index; power -1 is seen quite often for division.
there is a related problem (inspired by an interview questions from ITA software, a part of Google now) that asks to find the smallest positive integer that cannot be expressed using a digit K (in [2,9]) N times in an expression that involves the four basic arithmetic operations and the parentheses: +, -, *, /, (, and ).
to make the problem more interesting, the list of operations can be extended to include exponentiation and concatenation.
for example, using the four basic operations for N=4, and K in [2,9], we get:
7 cannot be computed in 4 2s;
11 cannot be computed in 4 3s;
10 cannot be computed in 4 4s;
8 cannot be computed in 4 5s;
9 cannot be computed in 4 6s;
4 cannot be computed in 4 7s;
5 cannot be computed in 4 8s;
4 cannot be computed in 4 9s;
i know everyone plays with their own rules, but from my perspective, a square root spends a 2 - it's not an operator like + or ! . there's an implicit "2" since you could also take the cube root or fourth root
You can use `sin(arctan())` as the repeatable operator instead of sqrt(). Then "unwrap" by applying ^-2 - (2^-2)
3=(sin(arctan(sin(arctan(sin(arctan(2))))))^−2 − (2)^−2
Perfect answer, I hate it :)
> it's not an operator like + or !
Squaring is to multiplication as is the latter to addition. If roots are invalid then so should be multiplications.
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 in the notation:
0 = 2 + 2 - 2 - 2
1 = 2 + 2 - 2 - 2 +
2 = 2 + 2 - 2 - 2 + +
Etc
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.
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.
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
Because it's just wrong. hahajk said that taking the 2nd root spends a 2, not that "roots are invalid". (FWIW, I'm not necessarily endorsing the view that the square root symbol implicitly spends a 2, I'm just saying that the counterarguments given are logically wrong ... actually more like semantically wrong.)
The comment below by drfloyd51 is similarly wrong ... hahajk didn't say anything about "giving exceptions to powers in the rules".
And the comment by speglt is also wrong:
> 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.
The number of operands isn't relevant; the degree of the root is.
And from SideQuark we get
> One could even claim division is raising to the -1 power, thereby disqualifying it in exactly the same manner you disqualify sqrt.
But hahajk said nothing about anything being disqualified. (Not to mention that division is not raising to the -1 power ... raising what? n/d = n * d^-1, but this is completely irrelevant to hahajk's argument.)
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.
Or use the successor operator.
Alternatively, one can claim sqrt is a single symbol, and there are extensions requiring a digit. Log, ln, also have hidden bases, but they’re also valid symbols. Cos/sin may be in radians or degrees, both in common use, and no real place to denote it.
One could even claim division is raising to the -1 power, thereby disqualifying it in exactly the same manner you disqualify sqrt. I’ve never seen sqrt with a 2 index; power -1 is seen quite often for division.
Go figure….
there is a related problem (inspired by an interview questions from ITA software, a part of Google now) that asks to find the smallest positive integer that cannot be expressed using a digit K (in [2,9]) N times in an expression that involves the four basic arithmetic operations and the parentheses: +, -, *, /, (, and ).
to make the problem more interesting, the list of operations can be extended to include exponentiation and concatenation.
for example, using the four basic operations for N=4, and K in [2,9], we get:
7 cannot be computed in 4 2s; 11 cannot be computed in 4 3s; 10 cannot be computed in 4 4s; 8 cannot be computed in 4 5s; 9 cannot be computed in 4 6s; 4 cannot be computed in 4 7s; 5 cannot be computed in 4 8s; 4 cannot be computed in 4 9s;
This feels very suspiciously like a finite field problem.
Are you allowed to brute force it?
That's how you do it right? It's a BFS with 4 being the max depth, and you print whatever numbers in the problem set which you didn't find.
Figuring that out is presumably part of the interview (and would depend on N)
22 is cheating because 22 = 2*10+2 is arbitrary in base 10. If you write 22 in some base, you have to write the base, and that's an extra number.
Popular last year (368 points, 165 comments) https://news.ycombinator.com/item?id=43145753
I'll just drop this here (yes, it's 4 instead of 2 - they're twice as good): https://fourmula.awsum.info/
Spoilsport.
And JFYI since December is near: https://rant.gulbrandsen.priv.no/hack/advent-calendar-for-ne...
Never understood why 7 is considered difficult...
7 = ⌈ 2.2² + 2 ⌉
Rounding spoils the fun of the puzzle.
It seems very much in the spirit of allowing concatenation and square roots...
Rounding is a completely valid mathematical operation.
(2 << 2) - ( 2 / 2)
It is difficult when the rules do not allow exponentiation.
There's a direct trivial solution for any natural number using the successor function S(a) = a + 1 from Peano arithmetic.
Encode 0 as 2 + 2 - 2 - 2 then apply S a times to that 0.
Only depends upon natural numbers & Peano stuff, no need for construction of rationals, irrationals, logarithms, etc.