## 2008-06-30

### Cedric's challenge, C port and analytic attempt

I was a bit disappointed that the C port of the previous code wasn't quite as fast as another approach which cached more state instead of generating the i'th value. The second attempt, using the bcd lists was comparable to John Wilson's Java version (1.3 seconds rather that 1.5 reported by the Java), and an attempt using bitsets a bit slower. Timings on Eee pc 900, Ubuntu hardy heron. The code is nrnat2.c.

The biggest jumps are between 9(descending digits) and 10(ascending) and 10(descending) and 12(ascending). Generating those four values will give the largest jump in a range.

The number of values will be the total of combinations: 9 + 9 * 9 + 9 * 9 * 8 ... up to the max value.

The total for 1..9 is 9*(9+1)/2 = 45;
For numbers 10 to 98 there are 9 numbers starting with each digit, so you have 450*9 for the first digit, and 9 lots of a odd sum, where there's one digit missing - so 9 * 45 - 45 for taking one digit out each time (the range is 0..9 instead of 1..9).

That gives a total up to 10 of 45

Up to 100 we get the 45 up to 10, 45 * 10 * 9 for the first digit shifted once repeated 9 times, and 40 * 9 for the second digit repeated 9 times (as the first digit is always non-zero, it biases the remaining digit's average by 0.5 from 4.5 to 4). So total for up to 100 is 45 + 45 * 10 * 9 + 40 * 9 = 4455.

Up to 1000 we get the values so far, 45 * 100 * 9 * 8 for the first digit shifted twice repeated 9 * 8 times - the number of combinations of the other digits. The remaining digits occur the same number of times as each other in each position and once for each value of the first digit, so you get 40 * 9 * 8 * 11 from them, so total is (4455 + 45 * 100 * 9 * 8 + 40 * 9 * 8 * 11) = 360135.

Similarly sum up to 10000 is (360135 + 45 * 1000 * 9 * 8 * 7 + 40 * 9 * 8 * 7 * 111) = 25277895

Simplifying, the sum of the values with n digits is (9!/(9-(n-1))!) * ( 45 * pow(10, n-1) + 40 * floor(pow(10, n-1) / 9) ) and the maximum jump is floor( ( 1203456789-1029876543 ) / pow(10, 10-n) )

Pete

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