# Three 4's Puzzler (11/7) has multiple answers (no spoilers here)

A friend an I have found 3 solutions to the 55=4&4&4 puzzler. I found one with an uncommon notation that is nonetheless known to all high schoolers. Then he found one with an obscure notation, but persevered until he found an elegant solution that is likely the intended answer. All 10th graders can understand this solution, so I encourage you all to try even if you’re not a geek (like some people I know).

So I asked my brilliant 15 year old daughter this question. She said “That’s easy, just add them up”. I said “That’s only 12”. She said “Turn it over!”.

This is the part where you are expected to post your solutions. If they really exist, that is.

Right. I had hoped they’d mention the alternatives on air.
Both of these use a viniculum (http://en.wikipedia.org/wiki/Vinculum_(symbol)), the bar over a repeating decimal, commonly taught in high school.

Mine: (4! + .4 viniculum)/(.4 viniculum)
That’s the same as 24.444444…/0.4444444…

My friend (who I’ll call Bill) used the Floor notation which is like square brackets with the top horizontal segments left off.

Bill’s: Floor( (.4 viniculum) / ( .4% (sqrt(4)) ) )
That’s the same as 0.444444…/0.008)

I struggle with the nature and meaning of existence, so I am not prepared to state that these solutions really exist.

Well done. I admit that’s a symbol I have never thought of in the context of this sort of puzzle. Huzzah!

You’re very kind. I had to look up the term for the repeating decimal notation, so I learned something new from all this. If asked last week, I would have guessed that “viniculum” was the black stuff under your fingernails from working on a car.

I wish I could edit out the multiple “Re’s” in the Subject line. This is starting to look like a Dorian mode solfege exercise.

Here’s mine. I sent it in as the answer and I was sure it was going to be the only answer, but I hadn’t considered that 0.4 would count as one of the 4s, but I guess .4 is alright. My solution uses what’s called “Divisor Function”, which adds up all the divisors of a number.

?(4) = 1 + 2 + 4
?(4) = 7
(http://www.deltaknoten.de/~bigben212/en/mathtools_divisor-aNA==.html)

Also used are the Factorial and the sqrt, as in the solution that was picked.

55 = 4! * ?4 + ?(4)
55 = 24 * 2 + 7

I agree that your solution is valid. It should probably get extra points for brevity and variety.

I think the challenge is most salient when letters (Roman or Greek or other) are excluded. I think the intention was to use the elite set of functions so common as to be represented by non-alphabetic symbols. Many possibilities surface when less-common functions are allowed–especially “nudging” functions that tend to move a number predictably in one direction. Nudging functions let you get closer to the solution by repeatedly applying the function. I would call the divisor function such a “nudging” function. Once it’s allowed, then multiple solutions appear.

I found one solution using the divisor function exclusively with the three 4’s (and an arithmetic symbol). I suspect there are others. Maybe it could be done with only two 4’s with the divisor function. I’m not going to try because I’ve obsessed too much already about this problem.

55 = ?(?(?(?(4)))) + ?(4 * 4)

I really don’t mean to diss your solution. I like all mathematical expressions. I didn’t even know there was a divisor function. I don’t want to be the grumpy amateur mathematician who fusses about petty details and doesn’t want marathoners to have a hearty 2500th anniversary party in 2010.

Just a quick follow-up on your follow-up:
55 = 4! + ?(4 * 4)

(4! - sqrt[4])/.4 = 55

Hmmmm. No complex notation necessary:

44 + 11 + 0/4 = 55

More complex notation is necessary if you stick to the rules. Only three 4s. Try again without the 11 and the 0. Maybe you can come up with something new.

I’m surprised how few valid solutions, other than the ones with the viniculum have been posted. I’m on board with the “rule” being to use only stuff that can be found on the normal keyboard, and no letters (i.e. no sigma, and my solution involving hyperbolic trigonometric functions would be considered invalid). Clearly using 11 or zero is wrong; the puzzler is not “three fours and as many other numbers as you want”. I humbly accept the restriction of no letters, although they don’t spell it out. Can we still use symbols like those for “floor” and “ceiling” that aren’t letters but don’t appear on the keyboard?

I still think that the sigma is alright. Because all it is is a function adding up the divisors of a number. Not much different than factorial, where you are multiplying all the numbers below n. The factorial sign (!) is just a short notation for this:
n! = ?k (with k=1 to n)
(http://en.wikipedia.org/wiki/Factorial)

I am not sure where you read / heard the rule of not using letters. Don’t think of it as “just” a letter. It’s the notation of a function, just as the ! for the factorial.

Anyway. Done with this puzzler, ready for the next.

I NEVER heard/read the rule of not using letters. I would normally be the one saying “they didn’t forbid it, therefore it’s allowed” which would allow my solution using hyperbolic functions, but they have never, to my knowledge, given a solution involving letters from any alphabet, so I have come around to the camp that says no letters.

Anyway, your notation was wrong. You need to specify the subscript, which means using other numbers; sigman sub 1 of 4 = 7, but sigma sub 0 of 4 = 3.

Yup. Right. And I am using the subscript on my page (http://www.deltaknoten.de/~bigben212/en/mathtools_divisor-aNA==.html)

Let me quote from wikipedia again:
“When x is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted, so ?(n) is equivalent to ?1(n)” (http://en.wikipedia.org/wiki/Divisor_function)

Fair enough. I saw that wikipedia had the sigma without the subscript, but I didn’t read the page carefully and thought I could sneak that by, just to win the argument. I still think letters aren’t intended in this type of Puzzler.