Talk:Puzzles/Arithmetical puzzles/Four 4s Equal.../Solution

What about the alleged "2" in "square"? The problem, as stated, explicitly allows the square operator, and using ^2 or superscript 2 is only one of a number of ways of expressing that operator. The square of X equals X * X. It's really the same with the square root operator: we may represent the root to be found with the normal root symbol, and a number, i.e., 2 for a square root, or 3 for a cube root, etc., but, routinely, the root is taken as the square root if another number is not specified.

If the square or square root is not to be used, the problem should state that. Instead, it states the opposite, so the claims of an IP editor that these are not allowed, because they contain a "two," are assertions involving a different problem. Many classical solutions found to the four fours problem involve the use of the square root operator, for some of the numbers, so it cannot be claimed that this is some deviation here. --Abd (talk) 20:56, 8 September 2010 (UTC)

Doing some more research on this, some sources -- including the Wikipedia article, don't allow the square, and some don't allow the square root on the same argument that would be used for the square. However, if we want to change the answers to not allow squares and/or square roots, the problem should be restated.

allows square root but not square.

Looking about, I see mostly the exclusion of the square operator, but, oddly enough, in one place, the allowance of the "sqrt" operator, which, of course, implies "2." I.e., it is equivalent to exponentiation at 1/2.

The problem is arbitrary, actually. The set of allowed operators could be smaller or larger. A smaller set makes the problem harder, perhaps. The source I cite did have solutions for all integers up to 116 without using the square operator, but using the square root.

claims that without using the square root operator, it is impossible to represent 10. However, our page shows an obvious counterexample for ten. It's a good example of how a little knowledge is a dangerous thing, and especially when "little" is "more than your average bear." Even a lot more. Experts make mistakes! --Abd (talk) 21:28, 8 September 2010 (UTC)

explicitly considers the problem. The author, David Wheeler, establishes a hierarchy of allowed operators. For reasons which escape me, he considers square root part of the basic operators, but lists square -- a very simple operation known to grade schoolers early on -- as number six. He calls this an "impurity factor," and, with the exception of square root, the operators he lists as basic don't imply some number. Percent, for example, impurity factor number 5, is division by 100. Before using square, he would prefer to use the overline, which, by the way, implies ten as a number base (as do some representations such as 44), an arbitrary root power (i.e., not merely two or four), the gamma function, i.e., (x-1)!, and percent. And he lists two more "impure" operators. Oddly, he seems to have a missing "impurity level," because the basic levels are called "zeroth," and he starts his numbering with level 2.


 * I did not necessarily agree with the IP editor who stated we can't use squares either. IMO, if squares are disallowed, sqrt should be too, and for the reasons Abd has stated.  Interestingly, the use of concatentation (i.e., 44) may or may not imply a base.  For example, the solution I added for 10 earlier today (44-4)/4 works for other bases as well, perhaps any base that can represent "4". (I know it's true for bases 5, 8, and 16, but the answer of "10" is in the selected base.  Maybe that disqualifies it.) This is an unsupported assertion, but it could be correct. It's another example of "little knowledge ;-)   Of course, that doesn't mean that concatenation works in other problems either.  All that being said, I do prefer solutions without sqrt or square, so I tried to find some. It's a fun exercise! --Jomegat (talk) 02:35, 9 September 2010 (UTC)
 * Well, no matter what the base, (44-4)/4 is equal to 11-1 or 10, that's true. Obviously this is any base 5 or higher. However, what we are looking for is to calculate an actual integer, the tenth item in the list of integers, with nine items before it! I have ten fingers, "ten" is just name for that integer. If the base is 5, using the digits 0-4, "10" is equal to 5 base ten. It doesn't work, it's the fifth integer, i.e., the one matching the fingers on one hand.... So using positional notation as in 44 does require a "number" -- the base -- to be specified. It's base 10 in all the examples I've seen. If we really wanted to be tricky, we'd require the base to be four! But would this use up one of our 4s? Heh! This gives us another way to use a 4 in the process.... Anyway, I found those IP comments to be a bit rude, so certain of being right, as if there couldn't be variant rules. --Abd (talk) 05:05, 9 September 2010 (UTC)
 * But if the base were 4, we couldn't use "4" anywhere else in the numbers, as 4 is not a valid digit in base 4. :-) I agree that the IPs comments were rude and that it was right to remove them. Thanks for doing so. --Jomegat (talk) 11:13, 9 September 2010 (UTC)

Square root operator
The problem allows the usage of the square operator and not the square root operator. Many revisions, including ones I worked on, overlooked this. Some users also used 44, contrary to another problem condition. (The problem is number base independent, to parse 44 requires assuming a number base, 10 was assumed, i.e., 44 = 4 x 10 + 4.) I have restored the answers given by the original author. I have not checked every one, that should still be done. --Abd (discuss • contribs) 13:18, 5 March 2014 (UTC)

The use of "44"
For one number, 43, I have been unable to find a solution that does not use either the square root function or the base-10 number "44."

The original statement of the problem did not disallow "44." This was added by this edit. All versions here did not allow the square root operator, but most versions of the solution allowed that use.

So we have

43 = 44 - 4/4, very simple, fits the original problem, without the restriction on the use of 44. I have it listed with a note that this is the only answer that requires 44.

There is another answer, 44, which is much simpler that what is given if 44 is allowed: 44 = 44 +4 -4.

Many other versions on the internet of this problem do allow the square root operator.

I was temped to do as had been done for quite a while, list 43 as "missing." Perhaps someone can think of a way to represent 43 without using 44. Or perhaps someone can prove that 44 is necessary (or the square root operator).

I personally prefer the problem as incompletely solved, it is pedagogically much more interesting that way. --Abd (discuss • contribs) 23:03, 5 March 2014 (UTC)