Does this puzzle have two valid solutions?

[Dahu]

I found this puzzle and sent it to a friend:

Using only the numbers 1, 3, 4, and 6, together with the operations +, -, ×, and ÷, and unlimited use of brackets, make the number 24. Each number must be used precisely once. Each operation may be used zero or more times. Decimal points are not allowed, nor is implicit use of base 10 by concatenating digits, as in 3 × (14 − 6).

The place where I found it says it has a unique solution

6 / (1-(3/4))

…but my friend came up with an alternative which was this:

6 x 4 x (1 x x 3)

He says that two multiplication signs in a row mean raise to the power. I don’t think I’ve come across this before. Is he right? Is this a legitimate solution?

[zombywoof]

@Dahu

He says that two multiplication signs in a row mean raise to the power. I don’t think I’ve come across this before.

Some computer programming languages (Fortran, Perl, Python are a few) use * for multiplication and ** for exponentiation.

[Peremensoe]

@Dahu

He says that two multiplication signs in a row mean raise to the power. I don’t think I’ve come across this before.

That works on some computers. You can make exponent calculations in Google’s calculator function this way, except using “*” rather than “x” (typing “2**3” yields “2 ** 3 = 8”). But I don’t think a mathematician would ever write anything like that, or accept it as a standard expression.

I give your friend a point for cleverness, but I wouldn’t say he’s formally solved it.

[borschevsky]

His second solution is a bit of a clever trick, but I would say that it isn’t a valid solution. The question talks about the operations +, -, x, /, and even though the string xx is made up of the character representing multiplication, exponentiation is still a separate operation.

If we’re allowed to use programming languages as justification, then my solution is:

6 x 4 // + 1 + 3

[Chronos]

Depending on the scope of what’s meant by “brackets”, you might not need any digits at all. For instance, 0 is {}, 1 is { {} }, 2 is { {} {{}} }, 3 is { {} {{}} {{}{{}}} }, and so on. 24 can be represented in the same way, though I’m not going to actually type it all out.

[Keeve]

Ditto what borschevsky said. If the puzzle had been worded “Using … the symbols +, -, ×, and ÷”, then your friend would have a good point. But it doesn’t say “symbols”, it says “operations”. And exponentiation is a different operation than multiplication.

So what you need to do is to use the word “symbols”, and change the “x” to “*”, and ask for TWO solutions, and you’ll have a great trick answer up your sleeve.

[nivlac]

As stated, the puzzle says to use the four operations +, -, ×, and ÷. That implies the standard use of these four algebraic operations. Exponentiation is a different operation. xx is not even exponentiation in computer lingo. It’s **. If you have an algebraic calculator, try punching in 6 x 4 x (1 x x 3) and see if you get 24. Of course not.