A problem that struck me while I was waiting in the incredibly slow queue to get coffee at work one morning, which I can't seem to get my head around - would be grateful for smarter mathematical minds than mine to give an opinion.
Scenario is: at work, I have a catering card. I top up this card with money every so often, and then I can use it to buy things (i.e. coffee) from the coffee bar.
Assume my card starts with £0.00 on it. Assume that, whenever I top up, I add £20.00 onto the card. Assume that I only ever buy coffee on the card, and coffee costs £2.40. Assume that if I do not have sufficient funds on the card to buy a coffee, I will add an additional £20.00 onto it. I will continue this cycle until my card returns to exactly £0.00. The card can never go into negative figures (so, if I have £2.39 on the card, I still need to top up with another £20.00 before I can buy a coffee).
Example: In this scenario, I would buy 8 coffees (leaving me £0.80 on the card), top up the card (so £20.80 on the card), buy another 8 coffees (leaves £1.60), top up (£21.60), buy 9 coffees and then finish (as card is now at £0.00). So my total coffee consumption, with cost-of-coffee at £2.40 and top-up-amount at £20.00, is 8+8+9 = 25 coffees.
The hypothesis is that,
for any pair of values* of cost-of-coffee and top-up-amount, the card will
always return to £0.00 at some point, even if it results in me drinking a near-infinite amount of coffee.
Or, to put it another way, there is no set of figures that would result in an infinite amount of coffee.
Is there a straightforward way to prove or disprove this hypothesis? It's been a few years since I stretched my mathematical muscles and I've bumped my head against this one a few times without making much progress. I suspect I'm missing something incredibly simple and somehow being a complete idiot, but I just can't work out how to start proving/disproving the above.
* The values have to be legal financial values i.e. two decimal place figures. I can't top up π pounds onto my card, nor can the coffee cost √2 pounds.