On 6/13/2012 10:35 AM, Stuart Anderson wrote:
https://fbcdn-sphotos-a.akamaihd.net/hphotos-ak-prn1/551099_429862577048001_...
there is a problem, what's the best fix?
My initial reaction was to want to make all the 9's the same size, thus I would change the 9-o-clock position to say (9x9)/9 or 9+9-9 or something like that. Using an exponent in the 1-o-clock spot would have the same problem. So instead, how about: sqrt(sqrt(9)/9) sqrt(sqrt(9)) where sqrt is square root; this simplifies to sqrt(3)/sqrt(3). On 6/13/12, George Hart <george@georgehart.com> wrote:
There are two problems in that image. Both are fixed here:
Wow… actually noticing that the clock doesn't keep good time… priceless!
On 6/13/2012 10:35 AM, Stuart Anderson wrote:
and how to generalise for all digits?
I would suggest the following "full table" method, because it's FUN and it's MATH!!! (-: In a spreadsheet program (or using paper like the good old days) make a grid with 3 rows labeled "one 9", "two 9's" and "three 9's", and columns for different values like the integers 1 through 12, but with extra space to add non-integers like "sqrt(sqrt(9))" if and when they come up. You don't have to keep all the columns in numerical order. In the first row, write "9" under the "9" column. Add "sqrt(9)" under the "3" column, then put ".99999..." in the "1" column (expressed as .9 with a bar over it). As you saw from my answer above, it's useful to make entries for things like "sqrt(sqrt(9))" which are not integers but might be useful later. If working with the digit "4", there would be a ".4" with a bar over it, which you'd make a new column for since it isn't one of the integers. In the second row, put "9/9" in the "1" column. We already have a way to make "1" from a single "9", but it's useful to have a way to make it from two 9's. Similarly, put "9/sqrt(9)" in the "3" column. Add other new formulas using two 9's in the appropriate columns. Use the first row to overcome writer's block: any combination of two things in the first row create a valid formula for the second row -- so you can just try all N(N-1)/2 pairings from the first row, and pair them in every way (A+B, A-B, B-A, AB, A/B, B/A, ...) So now we have "sqrt(9)+.9" for the "4" column and "9-sqrt(9)" for the "6" column, etc. For the third row, you can combine any one item from row 1 with any single item from row 2, OR you can use three items from row 1. As we know from looking at the clock pictures, everything through 12 is possible. However, a lot of clever thinking is still needed, I personally tend to overlook base-10-specific answers like "99/9" for 11 and "9.9 - .9" (with bars over the fractions). For a brute-force approach I use RIES, and I recently added options to make this sort of thing a bit easier. (Source code at mrob.com/ries; compile the source code and note the "--numeric-anagram" option. If you're interested let me know and I'll update the manual 8-) -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com