June 9th, 2006: The Google US Puzzle Championship is next Saturday, June 16th. It's used to select the US Puzzle Team, but puzzle-lovers of all sorts should find something interesting in the puzzles. Let's have a big "G" (for Google):

Here's a puzzle based on Erich Friedman's "Distance," from the 2003 Google U.S. Puzzle Championship. Label 14 of the circles above with different numbers from 1 to 14, such that the distance between 1 and 2 is less than the distance between 2 and 3, and so on. Four circles should remain empty.

If the "±" sign is selected, you can click on the top part of a circle to increase it and click on the bottom to decrease it. Or, select a number in the palette and click on a circle to fill that number into the circle. Deselect the active selection to create a blank. You win if all the "distances" to the right turn green.

Here is the solution to this puzzle, which was made difficult mainly because of the puzzle interface.

As with the first puzzle, I was wondering if there were other solutions to this.  Also how hard would it be to use a different letter? Like W, or O, or Q?

I believe there is only one possible solution.
I wrote a program in JavaScript (www.justwidgets.com/puz03_latest.html) to test all possibilities and it only finds one solution.

I'm not sure about other letters like "Q" or "O" but if you know HTML and JavaScript, I guess you could try changing the HTML table used by the program to see if it finds other solutions (keeping in mind the code assumes a square grid.
Distance puzzles with different shapes are here: www.stetson.edu/~efriedma/distance/

WOW, thanks!

No problem.
I created an improved version that lets you toogle the cells on and off with just a mouse click so you can try to create your own puzzle (No HTML/JavaScript knowledge required!) although the board is still fixed to 5x5. 

It probably wouldn't take much effort to allow people to change the number of steps required for the puzzle to be considered solved. For now, it's fixed at 13 steps, same as the original puzzle. With that many steps, it doesn't seem many letters will give solutions but I tried a few letters and found that "B" can work: