Larry the Looter

[Weehawk]

Recently, former Dual Nebulae Chief Referee Robert Kezcurm was playing a rousing game of Larry the Looter. In this fast-paced steal 'em up, one plays a ruffian named Larry who roams a city in chaos, robbing every poor individual he happens to encounter. For relieving a Girl Scout troop of their month's cookie inventory, the player scores 1300 points; for stealing a handicapped beggar's hard-earned spoils, the player scores 2300 points; and for overcoming the shotgun wielding hot dog vendor, one scores a whopping 7900 points. There is no other way to score points.

Between levels, Robert looked up at his score and noticed something rather striking. The first digit of his score happened to equal the number of 0's (zeroes) in his score. The second digit was equal to the number of 1's (ones) in his score. the third digit was the number of 2's (twos) in his score, and so on, all the way to the end, the nth digit in each case equalling the number of appearances of the digit (n-1) in his total score.

Given that the score rolls over at 10 million, what was Robert's score when he looked?

You are permitted to use a calculator, but are forbidden by the honor system to use any programming tool with logical operations.

For the first person to post the correct answer in the thread who does not already have one, I have a t-shirt:



If you already have one, you may PM me the answer and I will credit you in the thread.

[Weehawk]

Ross "VON" Benziger has answered correctly just for fun.

The t-shirt is still up for grabs.

[Milehighdt]

how about 3,211,000

[Weehawk]

how about 3,211,000

That would be it.

While it's obvious on inspection that this score fits the description, proving that it is the only possible answer is slightly more complicated. As a matter of fact I'll take one of the standard textbook copouts and say it is beyond the scope of this post. Suffice it to say for numbers of digits less than 10, here are the only numbers that fit:

n=1:   no sequence possible
n=2:   no sequence possible
n=3:   no sequence possible
n=4:   1210, 2020
n=5:   21200
n=6:   no sequence possible
n=7:   3211000
n=8:   42101000
n=9:   521001000
n=10: 6210001000

The question stated that the score rolls over at 10 million, so we know it wasn't 42,101,000; 521,001,000; or 6,210,001,000.

Why couldn't it have been 21,200 (or even 2020)? Those scores are not possible as any sum of multiples of 1300, 2300, and 7900. For analysis along these lines you may check this page on Frobenius Numbers. This also related to my question regarding Zantar II several years ago on the TG forum.

PM me your address and size for the shirt.