22

4-button method


14

I think the argument was over the question You may have said While your friend would have argued that there are indeed And won the argument by drawing something akin to the following line


10

$n=2^k$ button method Explanation:


10

They are trying to prove that How are they doing it? Will they succeed? A special case of the result being considered may be found at Maths SE (spoilers, obviously). The L-ish proof here uses the same underlying idea, but with the difference that


10

The circuit alone rigorously tests one ironic aphorism and another time-tested adage. A specific scientific hypothesis for the overall rebus: Interpretation The circuit has an input at the top whose components get copied and split to the right. The legend along the right of the picture identifies crossings as no-contact crossovers, square components ...


8

This is It can sometimes be applied Notes from this puzzle’s poser The column under $\small\underline{ ~ 1 - 9.\overline9 ~ }$ calculates... Calculations are logarithmic and have equal results across columns because... Such reasoning for logarithmicity is simple but adequate and the resultant skew in favor of I,X,C,M,... over V,L,D,... is ...


7

I think that Fiqo is trying to prove that: Like so: The reason Fiqo is not purebred is:


7

I think this is elucidating So, for instance, Perhaps more startling is


6

I believe this is the rearrangement inequality, perhaps with coin values. The rearrangement inequality is essentially saying that if you have two sets of positive numbers of the same size, and you pair one from each set with one from the other, take their products and sum, the greatest sum you can get is when you pair the greatest with the greatest, the ...


6

Addition plus geometry divides a game board Your friend’s graffiti motivation might have come from page 69 of Proofs Without Words, Volume 1, by Roger B. Nelsen, displaying an image exactly like a Connect Four game with a zigzag line instead of straight. Either way, a line can split the game’s cells (Thank you, wildBillMunson.) into two... Thus....


5

I think the configuration we are looking at is Here, courtesy of the OP, is a diagram showing how this works: Now, So diagram 1 is obtained by Then diagram 2


5

A different proof and a different friend, named Georg This is not the correct solution but does show, for variety, how a diagonal line in Connect Four resembles Georg Cantor’s diagonal proof that real numbers are uncountable.   So if your friend had been Georg, here is how his playing Connect Four might have meant more to him than just winning....


3

Maybe it had to do with


3

They try to prove Note that and are obvious reasons to fail. The proof can be elaborated like this:


3

The key thing seems to be that So perhaps I suppose I should say explicitly that It's not clear to me, though, As for the application in physics,


1

My solution is similar to noedne's. More bonus: Show that it is impossible if the number of buttons is not a power of two.


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