OddThinking

I’ve always liked this puzzle, ever since I read about it in Martin Gardner’s book, The Amazing Dr. Matrix.

I admire your effort to computerize the problem. I often do this for similar problems, but for the 4 4’s puzzle, I’ve never considered it. I guess I prefer to keep it as a thing to have for thinking about when driving or trying to fall asleep. (Two non-intersecting activities, I hasten to add.)

I had not assumed concatenation was legal. The initial statement of the problem says you are to calculate all integers from 0 to 100 using the number 4. Therefore, I thought it more reasonable to assume the postfix expressions to be evaluated were lists of tokens, with each token having one of seven possible values: 4, ‘+’, ‘-‘, ‘*’, ‘/’, ‘R’, and ‘!’.

Personally, I don’t like the idea of introducing operators that can only accept 4s as operands, but in trying to hunt down the “original” or “official” four fours problem, I found a four nines problem (attributed to Dudeney and paraphrased by me below) which was expressed in a way that makes concatenation of nines, and only nines, very clear and palatable:

Suppose you are labeling houses on a street using aluminum letters, digits, and other symbols you buy at a hardware store. Suppose the street has house numbers 1 through 100. Can you label every house on the street using a maximum of four aluminum 9s per house and any number of aluminum +s, -s, *s, /s, and parentheses? Letters may only be used to spell out the street name and no other digits or symbols are available.

Due to the context, it makes perfect sense that you should be able to simply put two aluminum 9s next to each other for house 99.

I like the idea of decimals and repeating decimals vastly less than I like the idea of concatenation (I am basically fine with exponentiation). While I do suspect the four fours problem with only the six original operators is insoluble, I would rather change the problem to “How many integers from 0 to 100 can you construct with a postfix expression consisting only of (etc.)?” Or perhaps a bigger range of integers, since 0 to 100 seems to be solved at seventy-something. Or “What is the fewest total number of 4s, coupled with these operations, needed to construct 0 to 100 (or number houses 0 to 100)?” Or how about allowing concatenation as the seventh operation, but make it a standard binary operation: When the concatenation token is encountered in the input, pop two values from the calculation stack, concatenate them, and push the result back onto the stack. Also, Wikipedia states that “most versions” of the four fours problem stipulate that you must use exactly four fours for each integer you are trying to construct.

Well, in mathematical terms, any “string operation” on numbers is just the same as multiplying by the radix to some power. Concatenation of single digits is just x ∙ b + y, where b is 10 in this problem.

I actually still dislike using concatenation to some extent, because it often requires fudging the exponent of the radix, so it doesn’t ever feel like an axiomatic operation. But the above approach is valuable for making “string operations” mathematically tractable, f.ex. when you have to prove something about numbers whose base n numerals show a particular relationship between particular digits.

(PS.: Julian, if you’re going to have mathematical discussion in comments, it would be very helpful to add <sup> and <sub> to the list of permitted tags. I rewrote my comment to avoid needing them, but that isn’t always possible.)

Aristotle,

^{Your wish} is _{my command.}

I wrote a quick WordPress plugin that added subscripts and superscripts, lists and definitions. I can’t see any security risks with the tags I’ve added (but I am wondering why they weren’t permitted in WordPress in the first place).

Still no image tags though; I am mulling over what sort of spam that might allow. No tables either, because that will take thinking, and I didn’t have the time at the moment.

Very nice!

Nit: it says <ins datetime="" cite=""> and <q>; I think you wanted to allow the cite attribute on the latter, not the former.

I don’t know why these tags are not permitted on a default install either. I can’t think of any effect either other than discouraging properly structured long comments – surely that’s not a goal? No images is just fine with me, btw, as is no tables. But lists are essential – and they’re allowed now, finally! The current lineup seems pretty complete in terms of common markup; off hand, I can’t think of any tags I might miss.

I’ve added cite to q tags. The cite attribute is allowed on ins tags. (Will anyone ever seriously use that tag in a comment? I doubt it!)

I would like images permitted in comments – e.g. the Man In The Moon article could have people submitting their own pictures. However, it leaves open the question of who hosts the pictures. Also, if I was a spammer, I would be actively seeking sites that allowed the posting of pictures in comments.

John,

Your arguments against concatenation and/or the rephrasing of the problem are all valid. It is funny that I just assumed its validity to begin with. It seemed natural to me.

(I didn’t even begin to consider the different bases, as Aristotle suggests.)

I share your distaste for the repeating decimals.

Aristotle,

A minor consideration when defining concatenation as a simple function – it is a function that only takes digits (or the results of the concatenation function) as an argument. You may not pass the result of another expression to it. This was something minor I had to handle when coding.

So what ever happened to this program? Well, it appears my software is still buggy, because the 9th cycle (i.e. working out formula that ar 9 digits long) took 2130 times longer than the 8th cycle, rather than the expected 8-10 times longer +/- 50% error bar due to other CPU loads.

I aborted it after the 9th cycle completed, and haven’t got back to it this week to work out what went wrong.

1=4/4
2=√4
3=√4+(4/4)
4=4
5=4+(4/4)
6=4+√4
7=4+4-(4/4)
8=4+4
9=4+4+(4/4)
10=4+4+√4

nerds

AGHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH help! lol. my teacher set this for my whole class to do as homework
when I went to school not long ago. I just thought that I’d refresh on all of this but there’s so much to remember, and the computer symbols are extremely confusing. I hated that teacher! Anyway I think that Katie’s comment may be wrong, as she doesn’t always use four fours which is the aim of the puzzle. Anyway, have fun working this horrible stuff out. xD

11. 4(2)-(4/4+4)

Jennifer,

Presumably by 4(2) you mean 4², which is not allowed.

44/4=11. (I am not adopting the “exactly four 4s” rule.)

Hi! I just tried out this problem on my YouTube channel, and continued it in a word document on my computer. I looked up the rules, and apparently, they’ve added 2 extra symbols that REALLY help. Along with exponents and the overbar, you can now use percents, AND, you can put a 4 in front of a √ to indicate the “4th root” of a number. Not only that, but you can put a .4 in front of the √, and that means the “.4th root”, which is just raising the number to the 2.5th power. Now I can get 32, 54, 100, and 120 with just 2 fours, and having benchmarks like this has helped me get to where I am now. I’ve gotten to 131 and plan to at least make it to 150.