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hubie [soylentnews.org] writes:

How can you guarantee a huge payout from any lottery? Take a cue from combinatorics, and perhaps gather a few wealthy pals [newscientist.com]:

By Jacob Aron [newscientist.com]
3 July 2025

I have a completely foolproof, 100-per-cent-guaranteed method for winning any lottery you like. If you follow my very simple method, you will absolutely win the maximum jackpot possible. There is just one teeny, tiny catch – you're going to need to already be a multimillionaire, or at least have a lot of rich friends.

[...] Picking numbers from an unordered set, as with a lottery, is an example of an "n choose k" problem, where n is the total number of objects we can choose from (69 in the case of the white Powerball numbers) and k is the number of objects we want to pick from that set. Crucially, because you can't repeat the white numbers, these choices are made "without replacement" – as each winning numbered ball is selected for the lottery, it doesn't go back into the pool of available choices.

Mathematicians have a handy formula for calculating the number of possible results of an n choose k problem: n! / (k! × (n – k)!). If you've not encountered it before, a mathematical "!" doesn't mean we're very excited – it's a symbol that stands for the factorial of a number, which is simply the number you get when you multiply a whole number, or integer, by all of those smaller than itself. For example, 3! = 3 × 2 × 1 = 6.

[For the US Powerball lottery] Plugging in 69 for n and 5 for k, we get a total of 11,238,513. That's quite a lot of possible lottery tickets, but as we will see later on, perhaps not enough. This is where the red Powerball comes in – it essentially means you are playing two lotteries at once and must win both for the largest prize. This makes it a lot harder to win. If you just simply added a sixth white ball, you'd have a total of 119,877,472 possibilities. But because there are 26 possibilities for red balls, we multiply the combinations of the white balls by 26 to get a total of 292,201,338 – much higher.

Ok, so we have just over 292 million possible Powerball tickets. Now, here comes the trick to always winning – you simply buy every possible ticket. Simple maybe isn't quite the right word here, given the logistics involved, and most importantly, with tickets costing $2 apiece, you will need to have over half a billion dollars on hand.

[...] One of the first examples of this kind of lottery busting involved the writer and philosopher Voltaire. Together with Charles Marie de La Condamine, a mathematician, he formed a syndicate to buy all the tickets in a lottery linked to French government debt. Exactly how he went about this is murky and there is some suggestion of skullduggery [laphamsquarterly.org], such as not having to pay full price for the tickets, but the upshot is that the syndicate appears to have won repeatedly before the authorities shut the lottery down in 1730. Writing about it later, in the third person, Voltaire said "winning lots were paid in cash and all in such a way that any group of people who had bought all the tickets stood to win a million francs. Voltaire entered into association with numerous company and struck lucky."

[...] Despite the fact that the risks of a poorly designed lottery should now be well understood, these incidents may still be occurring. One extraordinary potential example came in 2023, when a syndicate won a $95 million jackpot in the Texas State Lottery. The Texas lottery is 54 choose 6, a total of 25,827,165 combinations, and tickets cost $1 each, making this a worthwhile enterprise – but the syndicate may have had assistance from the lottery organisers themselves. While the fallout from the scandal is still unfolding [nytimes.com], and it is not known whether anything illegal has occurred, the European-based syndicate, working through local retailers, may have acquired ticket-printing terminals from the organisers of the Texas lottery, allowing it to purchase the necessary tickets and smooth over the logistics. [...]

So there you have it. Provided that you have a large sum of upfront cash, and can find a lottery where the organisers have failed to do their due diligence with the n choose k formula, you can make a tidy profit. Good luck!

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