My Google personalised homepage has a puzzle on it that has me completely stumped. The question is as follows:
Any body got a clue how to do it? I don’t think there is a prize or anything, it’s just a bit of fun, but it’s driving me crazy!!
Wow… that’s a toughie!! I actually used my programming skills to solve it systematically… there are 5 possible unambiguous patterns of parenthesization you can use on an expression of 4 values, 6 different orders in which you can use the 8’s and threes, and 4 choices for each of the 3 different operators. That’s a total of 1920 different possible expressions you can build using the rules - I wrote a VB program to generate them all, copied them into OpenOffice Calc, let it evaluate them all, and searched for the one with a value equal to 24. It’s very tricky, involving fractional math at every step… and I can’t think of any better clue to put you on the right track without giving it away.
Answer in spoilers:
8 / (3 - (8 / 3))
8 divided by three is 2 and two thirds
3 subtract 2 and two thirds is one-third
8 divided by one third is the same as eight times three, or 24.
I managed to find a solution rather quickly by working partly forwards and partly backwards. First, the forward reasoning: Using +,-,/,* the only values you can make from two 3’s are {0,1,6,9}. Similarly, two 8’s can make {0,1,16,64}, and from one of each you can make {-5,3/8,8/3,5,11}. Now, see whether a solution is possible of the form (_ op ) op ( op _). It is simple to check that there is no way to combine one value from {0,1,6,9} with one from {0,1,16,64} to make 24. Also it is not possible to combine any two elements from {-5,3/8,8/3,5,11} to make 24.
Therefore that symmetric form is not possible, which means that the outermost operator must have 3 or 8 as one of its arguments, resulting in a value of 24. Now we work backwards, and list all the possibilities:
(21) + 3
(27) - 3
(8) * 3
(72) / 3
3 - (-21)
3 / (1/8)
(I’ve only listed those involving 3, but you get the idea)
Now the problem is to find how to make one of the values in parentheses, using 3 or 8 together with one of the numbers in braces above. There really aren’t that many combinations to check. I found my solution in less time than it took to type up this response.
Here’s a hint:
Try to make 8 / (1/3).
Here’s the solution I found:
8 / (3 - (8 / 3))