Math Fun

Still missing:
8 8 8=6
9 9 9=6

The rest:
tab!=6
(1+1+1)!=6
2+2+2=6
(3x3)-3=6
(4-(4/4))!=6
5+(5/5)=6
6+6-6=6
7-(7/7)=6[/tab]

I think I need to incorporate another operation…

Good effort! :slight_smile:

8s are the hardest, btw.

[tab]Good job thinking of using factorials - that was the least obvious one to me.

You’re right about needing to incorporate another operation to finish things off.[/tab]
Don’t open the next tab if you want to see if you can do it all without any help…
[tab]Here’s a clue, squirt. Another operation would neaten up the 4s a little while you’re at it.[/tab]

I figured out a cheat-y way that solves all of them:
[tab]( ( d/dx(8) )! + ( d/dx(8) )! + ( d/dx(8) )! )!

That works no matter what number you use, because the derivative of any constant is 0, and the factorial of 0 is 1. It’s a bit of a cheat because d/dx is arguably adding another number (dx).

It also makes the problem boring.

Still no legit solution to 8 and 9.[/tab]

Haha!

Wow, it’s amazing the creativity that difficult problems can drive us to.
The function you’re looking for really is fairly elementary.
Another hint, should you need it:
[tab]Start with the 9s. Think about what you can do to a number like that, especially in light of a number I’ve already mentioned in my previous hint.[/tab]

Hmmm…
[tab]I think you’re suggesting taking the square root, which I wrote off because I saw it as incorporating another number: 2.

If that’s what you’re suggesting, 9 is easy: ( 9/sqrt(9) ) +sqrt(9) = 6
and 8 - sqrt( sqrt( 8+8 ) = 6
But why not cube root + cube root + cube root?
And if we’re doing that, why not x root of c + c + c, where x is whatever irrational number root results in 6?
Or log base y where y is a number that when raised to the 6th power equals c + c + c?

I don’t know why I treated the square root as adding a number, but not the factorial. As I think about it, I find the line between adding another number and just adding an operator to be fuzzy.
After all, you can always define a new operator. So we could define the unary operator B such that Bc=(c x 0) + 6.
B(c + c + c) = 6 for all c.[/tab]

[tab]I consider square root to have a hidden number as well, Carleas. It’s raising to the power of 1/2. Just because we happen to have a symbol that doens’t look like a number that means ‘raise to the power of 1/2’ doesn’t mean it’s not using a number. I mean, I could make a symbol that means Raise to the power of 0 and multiply by 2…right? I could just invent that symbol. Maybe it could be the dollar sign. 8£ + 8£ + 8£ = 6. BOOM! Realistically, that’s just as valid as using square root.[/tab]

The point to the puzzle was to stick to conventional mathematics as it currently is with its current symbols. Adding new creative “operators” isn’t acceptable.

[tab]Yes, I was suggesting that you use the square root, since it can be denoted by √ which does not require using a number. Contrast this with other roots such as the cubed root that requires using the number “3” however you present it. The square root may also be represented by raising a quantity to the power “1/2”, which involves using a number, but using the √ symbol remains within the designated rules. The ! symbol to denote “factorial” is fine for the same reason, even if its expansion would involve adding other numbers.

Perhaps I didn’t explain that rule sufficiently.

Also, only the integers provided at the beginning of the problem are the only ones allowed - so no “x root of c + c + c, where x is whatever irrational number root results in 6” or “log base y where y is a number that when raised to the 6th power equals c + c + c”. And no use-defined operators either! Lol. Only conventional mathematical ones - I should have been more explicit on that one too :stuck_out_tongue: Nice idea though. I’ve enjoyed all of your contributions to this problem, even though some of them did not fall inside the rules I was trying to communicate.

( 9/sqrt(9) ) +sqrt(9) = 6 and 8 - sqrt( sqrt( 8+8 ) = 6 are both lovely.

I would even say they’re better than the solutions I was expecting, which were:
(√9 x √9) - √9 = 6 and (√(8/8 + 8 ))! = 6

Though I do prefer √4 + √4 + √4 = 6 for the 4s.[/tab]
So Carleas has solved them all =D>

A shame nobody else contributed.

EDIT:
I got distracted writing this reply, as usual, so by the time I previewed it there were more contributions.
JSS, that’s correct.
FJ:
[tab]By that argument, all conventional mathematical operators would involve extra numbers if fully expanded. My bad for not explicitly stating exclusively conventional mathematical operators in the rules. Carleas has already experimented with user-defined operators in much the same way as you suggest, though maybe you resisted opening the tabs which show this.

Not to say both of your arguments are invalid - they’re good points. I just didn’t mean to imply that you could use them when I described the problem. Both square roots and factorials were supposed to be fine, along with addition, subtraction, multiplication and division. No numbers have to be used when using certain conventional mathematical operators for each of these.[/tab]

I came to this problem late but I will say I did like the way Carleas called differentials a cheat, I don’t think they were since they don’t use numbers per se but assume general solutions to the differences between one value and another with a related dependant value and rule to evaluate it. In fact every single one of the values could be solved by differentials, substitution rules etc, addition of one number is irrelevant if you use the right maths, in fact you can basically solve an value of summation in a differential without an operator other than d/dx, it’s kinda too easy though if you know A’ level plus maths, and you can see why it would not be allowed. Kudos to the solutions that weren’t “cheats”. :slight_smile:

I’ve got a ‘puzzle’ so to speak, but it’s pretty open ended.

I’ve got a data set: I’ve tested drug A and drug B on both women and men. For the sake of simplicity, we’ll just say that each subject either died or healed (just a binary measurement of the effect of the drug on each individual).

The percentage of men who were healed by drug A was higher than the percentage of men who were healed by drug B.
The percentage of women who were healed by drug A was higher than the percentage of women who were healed by drug B.

But, the percentage of people overall who were healed by drug B was higher than the percentage of people healed by drug A.

How is this possible? What does a data set have to look like to produce this result?

And no, it’s not a trick question – the answer is not anything like “Many of the participants were hermaphrodites or transvestites.” Just keep it simple: all participants are men or women.

Bonus points if you know the name of this mathematical anomaly.

I spent a while on this before I gave up and googled it, and learned that I had added some assumptions that were getting in my way.

This first entabment will only discuss assumptions, but I tab them because talking about them can sometimes suggest the answer; purists may want to avoid:
[tab]I had assumed that the treatment group sizes were the same, so that the number of individuals given each drug were the same, and thus that the number of men given drug A was the same as the number of men given drug B, and ditto for women. The trial sizes and compositions can be different.[/tab]

This next entabment will discuss the answer:
[tab]I’d never heard of Simpson’s paradox before this, so thanks for pointing it out. They have some interesting real-world examples in that article. Particularly interesting was the example of gender-biased admissions.

I’m having difficulty succinctly expressing what must be true of the data set for this to occur… I know that the compositions need to be reversed, so for example, more women participate in the drug A trial and more men participate in the drug B trial. And I intuitively grasp the “vector interpretation” given in the wikipedia article, especially the image:

But I need to think more on this to distil what’s going on. Every time I try to express the criteria that are necessary, it sounds like a tautological restatement of the question itself. I know the two trial groups must be of different sizes and ratios of men to women, and I think it is expressible as some relationship between the success group of one drug for one gender to the failure group of the other drug for the other gender.[/tab]

[tab]Carleas: That’s the one

I would explain it like this:
It’s about weighting (think weighted averages).

Women have a higher healing rate than men. A has a higher healing rate than B.
If the healing rate of Women taking B is higher than the healing rate of Men taking A, and A is disproportionately taken by men while B is disproportionately taken by women, this (with some more mathematical constraints) can allow for the Simpson’s Paradox to occur.

I’ve taken a screenshot of a journal article about it that you might find enlightening:

[/tab]

A new one circulating the net, similar to other problems that have been discussed here.

[tab]August 14

Edit; lol no, that can’t be it… has to be either June 18 or July 16

Edit2: July 16[/tab]

Yep.

[tab]I’m curious why you said it has to be June 18 or July 16. I eliminated May and June first and went from there, I’d be interested to know your reasoning.[/tab]

Like most of these types of things it is poorly worded.
Fact is that either having the knowledge but not the other, alone is simply not enough to choose between the the dates and the 3 line conversation is of no help.

Unless we know the reasons why the 3 line conversation occurs - such as how they know then the problem is insoluble given the information.

There was a similar problem in this site a couple of years ago, about Monks that was equally stupid.

May as well post my explanation

[tab]I’m wasn’t looking very carefully or focusing on it much because of work, so yeah, June 18 makes no sense :slight_smile:

Albert: there are only 2 daya that don’t repeat, and they don’t match the month I know, so Bernard can’t know the answer.

Bernard: if Albert knows that I don’t know the date, then it can’t be May or June, because those months have the only numbers that don’t repeat. Thus, since I know the day, and Albert ruled out May, it is July 16.

But now that I think of it, why not August 15?

Edit: because Albert then says that he knows it too. The criteria above eliminate all days except for July 16, August 15, and August 17. If Albert now knows it too, it can’t be August, since there are still 2 dates to choose from.[/tab]

This solution does not work.

Oh, I got it… I’ll edit my tab above.

[tab]However. B knowing that answer can ONLY conclude that June the 18th is the day, as he knows it is the 18th and from the list he can see only one date with that number. On hearing that B knows, A can also conclude that the birthday has to be on the only date that has an exclusive day number. As of all the dates, only one month has a day number not shared with any other month.[/tab]