How amazingly beautiful are those patterns, don’t you think? And how about those proofs?
Faaantastic :) Anyway, those two patterns look very similar and the two proofs were also pretty
much identical. Now, doesn’t this suggest that these patterns form the start of an infinite
sequence of similar infinite patterns? An infinite pattern of infinite patterns. That’d be cool,
right? Just adding up consecutive integers on top. Then squares. Then what? Cubes of course! What
do you think? Is this going to work? Well, let’s have a closer look. What would the first equation
for the cubes pattern look like? Well, obviously, something like this: A cubed plus B cubed equals C
cubed with A, B, and C being positive integers. Hmm, actually that doesn’t look good at all? As
ALL Mathologer regulars will know this is actually impossible. Fermat’s magafamous last theorem named
after the 17th century mathematicial superstar Pierre de Fermat tells us that this equation
does not have any positive integer solutions A, B, C. A cubed plus B cubed equals C cubed with A,
B, and C being positive integers. So there won’t be any nice pattern for
cubes. In hindsight maybe not that surprising. Remember, what made things work, for example,
for the squares was this ingenious reassembly of a square into a shell for a larger square.
The counterpart for cubes just doesn’t work out. There, for the nifty construction to work in 3d
we’d expect having to create six equal slices but those six slices don’t fit nicely around
the six faces of a smaller cube. Anyway, cubes are out. And, actually, much more than cubes is out.
Fermat’s last theorem also says that even if we replace the 3s in the exponents by any other
integer n greater than 2 the equation A^n+B^n=C^n won’t have any solution with positive integers
A,B,C. Hmm, so there goes our sweet dream of an infinite pattern of infinite pattern. Sad :( On
the bright side, not having that mega pattern makes our two patterns extra special. There,
EXTRA special. Actually there is a way to make both patterns even more special. I’ve known
about these patterns for most of my life and I always felt that there was something missing at
the top of both patterns. Right? Shouldn’t there be something there? Well, while preparing for
this video I had another really close look and I came to the conclusion that it would make sense to
extend the patterns like this. So 0 equals nothing at the top of the first pattern and 0 squared
equals nothing at the top of the second pattern. Bit weird but don’t the patterns now look much
more complete? Aaand …. Behold, two very festive looking trees just in time for Christmas :) To
finish things off for today, here are three nice puzzles for you to ponder. Each of these puzzles
hides a nice AHA moment and so don’t skip them :) First puzzle: if the cubical pattern actually
had worked out, what would the first A cubed + B cubed = C cubed equation have been? Okay, okay,
we already know that the two sides of that would be “equation” won’t be equal. However,
what exactly is the difference between the two sides? Nice little surprise waiting for you
there :) Record your answers in the comments. Second puzzle: Over there is the 1895 painting
by NIKOLAY BOGDANOV-BELSKY: entitled “Mental Arithmetic. In the Public School of S. Rachinsky.”
Can you solve the problem on the board in your head in under 30 seconds? What’s the answer? And
what’s your strategy for attacking this problem? Again, leave your answers in the comments.
Final puzzle: 3 squared plus 4 squared equals 5 squared. How about 3 cubed
plus 4 cubed plus 5 cubed equals what cubed? And what about the next sum in
this sequence? Be surprised and leave your answers in the comments. Anyway,
that’s all for today. I hope you enjoyed this somewhat unusual Mathologer video.