double numbers

[gearhead]

Are there any levels that have double numbers on Operation Gumball?
For example XX22X or 7XXX7.

How about triple numbers or more?
For example 99XX9.

[jillion]

Yes. There are 30 Levels, the last 20 of which allow duplicate digits. Or as williamson_39 says elsewhere:

Originally Posted by williamson_39

30. There are three sets of ten. Each set has codes of 3, 3, 3, 4, 4, 4, 5, 5, 5, 6 digits long (but you get fewer chances each time).

In levels 1-10, there are no duplicate digits in the codes, but duplicate digits appear in levels 11-20 and 21-30, which also make the higher levels more difficult.

In fact, sometimes the the entire string of digits is all the same! Always surprises me when it occurs. Even though I generally only play one nightly game (when I remember to play), I've seen this more than once. See pic below for one such occurrence.



p.s. I am tempted to derive the actual proportions of strings (i.e., codes) that have duplicate digits and compare the number with those that do not (for Levels 11-30), but realize this would be me procrastinating my other more pressing work! LOL. This may be done elsewhere anyway, somewhere in WI.

At Level 11, for instance, only 280 of the possible 1000 strings have at least one duplicate, whereas 720 of them do not have any duplicates. So, 72% of the time Levels 11, 12, 13 and 21, 22, 23 will not have any duplicate letters in those 3-digit strings (assuming Ganz's programmers allow each possible outcome to occur with an equal probability).

I would love to derive this result, here, so that everyone can work it out for themselves and see the simple reasoning involved, but will not in the interest of staying on my other tasks. Perhaps later, if anyone is interested in a brief introduction to probability, counting, and exploring the sample spaces of random events. It is actually very cool to think about and not complicated.

[williamson_39]

Originally Posted by jillion

...At Level 11, for instance, only 280 of the possible 1000 strings have at least one duplicate, whereas 720 of them do not have any duplicates. So, 72% of the time Levels 11, 12, 13 and 21, 22, 23 will not have any duplicate letters in those 3-digit strings (assuming Ganz's programmers allow each possible outcome to occur with an equal probability).

I have never thought about it in these terms - how unlikely it is to have duplicates in the 3-digit levels. Very cool! I had noticed that I rarely saw duplicates in the first few levels of the second set (11-20), and wondered if the "allow duplicates" change was really happening at level 11, or not until a little later. But this makes perfect sense. Brilliant, jillion.

Thanks for such a cool answer. I hope you do share more of your insights. I love the way you shine your light on these topics - always a pleasure.

[webkinzz x3 <<<loveee.]

there are diff. levelss
kay sweet

[jillion]

Originally Posted by williamson_39

... But this makes perfect sense. Brilliant, jillion.

Thanks for such a cool answer. I hope you do share more of your insights. I love the way you shine your light on these topics - always a pleasure.

Pleasure's all mine, my friend. I do take such great pleasure in thinking. Now, williamson_39, could you build me some sort of time-machine so I will have more time to share my insights, please? Or perhaps an anti-procrastination machine?

Actually, deriving the results and explaining them (as I mentioned I would like to do above), may fit right in with my current tasks: I'm gearing up to work on a CME project (Center for Mathematics Education) where the intended audience is high schoolers, unlike my usual audience of college-level students or other mathematicians.

For now, I will leave the derivation to the interested reader, as is so often said by us mathematicians.

p.s. Good Fortune would have it that the two chapters I will be putting the final polish on for the CME collaborative text are the best two: Combinatorics (counting, etc.) and Probability. Lucky me! I'm fortunate enough to get paid for doing what I love. May everyone be so lucky!

[chumpkin]

I think the game is random. Many times it doesn't have duplicate numbers even when you are above level 10. I saw triple numbers only once.

[kirsten012]

oh my goodness! these levels are so hard!!

[jehannah4]

Good luck on that CME project. I'd give you an A+!

[jillion]

Originally Posted by jehannah4

Good luck on that CME project. I'd give you an A+!

Thanks, jehannah4, for wishing me good luck! Chance would have it that my project was delayed a bit, so I have just started this morning on the combinatorics chapter (which presents the art of counting without counting).

I was coming here to WI to procrastinate a bit, actually. But as luck would have it, your post has inspired me to get back to work!!! LOL. Thanks so much, most excelllent timing!