Popular last year (368 points, 165 comments) https://news.ycombinator.com/item?id=43145753

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 ⌉

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It is difficult when the rules do not allow exponentiation.

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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)

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;

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.

Given the problems ITA solved, thinking about the question that way makes perfect sense.

Figuring that out is presumably part of the interview (and would depend on N)

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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

> 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.

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.

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….

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 :)

There's a 2 in the definition of sin :-)

Though it really depends which definitions you find more fundamental.

Still, exp and log seem like good choices.

> There's a 2 in the definition of sin :-)

How?

sin(x) = (e^ix - e^-ix) / 2i

That's more an identity than a definition.

Keep it simple.

2^2 - (2 / 2).

2 + 2 - (2/2).

Or use the successor operator.

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.

This and Dirac's trick are both writing the number in unary. Is there a way to write it with o(n) symbols?

That's a fun question.

If we're allowed to introduce our own operators, we could concoct something like decimal notation

a . b := 10a + b

So we could write 90210 as 9 . 0 . 2 . 1 . 0 = S(S(S(S(S(S(S(S(S(0))))))))) . 0 . S(S(0)) . S(0) . 0

That doesn't quite work as it still depends upon a bunch of literal zeros. We could replace 0 with an operator Z() := 0, but that seems to be very much against the spirit of the challenge.

We could rewrite our ersatz decimal expression to only require a single zero & then evaluate a vector of functions f : N -> N across that zero, resulting in a vector of natural numbers, then reduce that vector to a single number with our `.` decimal notation operator.

Let M(fs, x) := (fs_1(x), ..., fs_n(x))

Let R(x) := x_1 . x_2 . ... . x_n

90210 = R(M((SSSSSSSSS, Id, SS, S, Id), 0))

the number of symbols to write a d-digit number this way is bounded by 10 + 10d = 10 + 10log_10(n)

Is this cheating / against the spirit of the challenge? Perhaps!

I'll just drop this here (yes, it's 4 instead of 2 - they're twice as good): https://fourmula.awsum.info/

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You don't even need zero in that case, you can use the fact that 2 - 2 = 0 to use the four 2s:

  S(S(S(S(S(S(S(2+2-2+2)))))))) = 7

...which is essentially the same as Dirac's general solution (which also derives a "successor operator" akin to S that doesn't spend any numeric literals).

EDIT: I realize that this was already posted in a comment by shoo several hours ago.

You’ve got S(4) in the center there

From the text, the inner term was clearly intended to be "2-2+2-2".

I agree!

A recent IOCCC entry solves a related game of making integers out of variable number of constants (e.g. make expression using only "2" and basic operations, but any number of "2" can be used as opposed to always four):

https://www.ioccc.org/2025/kurdyukov/index.html

I had a teacher in middle school who would assign a version of this over winter break with the digits of the new year. I had a lot of fun with it in 1996. Sad to think she must've stopped in 2000...