1,3,14,22 : 1 * 3 * (22-14)
1,2,3,4,5,6
1, 5, 5, 5
5 * ( 5 - 1 / 5) = 24
I hope a little self-promotion is allowed after finishing such a challenging puzzle ;-)
Everything you want to know about math 24 ----------> Play math 24 game - 24theory, solves 24 the math game
Here is a hard one
1 4 5 6
Give TWO solutions
If you need help ---> Play math 24 game - 24theory, solves 24 the math game
1,1,2,8
(1+2) * 8 * 1
That's the only solution. 8 * (2 + 1) * 1 is the same, ok? Read my little article on this issue ---> The theory - 24theory, solves 24 the math game
Sorry for the self-promotion. If anyone is offended, let me know.
[ 5 7 7 11]
So you are saying (1 + 2) / (1/8) and (1+2) * 8 * 1 are two different solutions. But I disagree. If you read the "theory" part of the website I put up. I explained the methodology of determining the so called "unique" solutions. In short, it is aimed to get rid of the gimmick "new" solutions.
(a + b) * c * 1 is always equal to (a + b) * c / 1, correct? (or you can say the second solution is a gimmick)
And (a + b) * c / d is always equal to (a + b) / (d/c), correct?
So let d be 1, we always have (a + b) * c * 1 = (a + b ) / (1/c).
They are the same thing for any a, b and c. So they are essentially the same solution.
Hope it's clear.
THIS thread was started by OP Mrdavid445 who did not state the conditions you're "making up".
These 2 solutions: (1 + 2) / (1/8) and (1+2) * 8 * 1 would be considered different: one uses *, one uses /
I suggest you should have started your own thread, with YOUR conditions.
I did not look at your "website on this"; also I am not saying you are "wrong"; I SEE what you're getting at.
As example, one "manner" is to look at this style: u / (v - w/(w+1)) = 24.
Then v = (24w + u(w + 1)) / (24(w + 1))
Easy to loop through (keeping all 4 numbers < 100) to get solutions:
here, there are 31 solutions: from 1 / (1 - 23/24) to 99 / (5 - 7/8)
so a + b + c - d and a + b - (d -c) are two different solutions, or not?
The point is, one has to draw a line somewhere to get rid of the gimmicks. I happen to believe a * b * c * 1 and a * b / (1/c) are the same solution. But that's just my opinion. We can live with different standards.
And I don't understand the second part of your post. Does it have anything to do with our discussion of the different solutions or you are just pointing out a way to find interesting puzzles?
You're missing my point: what one believes here is NOT important.
Simply state your belief as a RULE in the puzzle.
Once more: I posted (1 + 2) / (1/8) and (1+2) * 8 * 1 as 2 solutions
because the ORIGINAL wording of the puzzle did NOT prohibit that.
If YOU want to prohibit that, that's OK: but start a NEW thread and SAY SO.