The Moessner Miracle. Why wasn't this discovered for over 2000 years?

The sum of these would be the number of ways to get to the orange circle and of course

this

is exactly how the pascal numbers are generated so that the travel route numbers start the same way and accumulate. in the same way as the pascal numbers, so they must always be equal to the pascal numbers, easy, what about the other triangles? How well are they built? Let's take a look at twisting the second triangle so that it remains a Pascals triangle. The numbers on the left edge are no longer all ones, but the numbers in the middle are obtained by the same addition rule: 6 plus 1 equals 7. 16 plus 7 equals 23 and so on, and now that we know What we know, it's not hard to check that not only do the numbers in the first triangle count the number of trips from the blue circle but, in fact, all the numbers you see here count the number of trips in

this

larger arrow diagram ?

Take my word for it or convince yourself that this is really easy. anyway as an example to get to this circle label 6 there we have to go through one of these two circles and of course five plus one and six works another one down there one plus 31 is 32 and another 32 plus 211 is 243 and so on in super Duper Nifty Now comes the key trick: Reverse all the arrows Now all the reverse parts are heading to the target blue circle at the top So what do the numbers indicate now? They tell us the number of trips from that circle to the target circle at the top.

Well, the stage is set. Remember that we are trying to show that most addition methods result in successive fifth powers. in the lower circles of the triangles, then what we have to prove is that from here it is exactly one trip to the blue target, which is pretty obvious. So we have to show that from here there are 32 trips to the target and that from here there are 243. and so on, well, let's start from the red circle, then it is completely obvious that there is only one way to get from the red circle to any of the top right circles, let's move the red circle to the right, okay, how about the number of trips from this red circle? from the circle to the blue target circle, let's count them from zero.

Well, first going up, that's all once again. What's up with this circle here? Well, there is only one arrow that leads to it and it originates from one. This means that the circle in questions also has to be one in other words there is only one trip from red to this new circle and obviously the same is true for all these circles now one plus one is two the same for all these circles alright two plus two is four and therefore the next diagonal is the four to four, the next one is eight 16 32 and there we have the 32 that we were looking for and a very good power pattern of two in the triangle on the left to get there , you can feel how this works, let's start with the next card, so because of the repeating pattern, our counting starts again like this, nice, eh, arguing like before, we got this number transfer now, let's fill in the remainder 243 in the blue circle as shown I hoped, great, but we'll get to the explanation in a moment, I promise. just do one more count first there, like before we get this number transfer and so on to spot some patterns, for this we will scan things from right to left, notice the powers everywhere you look, let's go there again, the powers of a turn. in powers of two the powers of two become powers of three and so on, very very nice how all the pieces fit into place so it has to be correct if we can prove this pattern that in general the powers of n are on the right side of a triangle become the powers of n plus one on the left side of the triangle, then we will have shown that the

miracle

mercenaries have a clear hope, so okay, to find out what is going on, let's see how the powers of three on the right side combine into the powers of four on the left let's focus on this entry here, obviously the only powers of three on the right that can contribute to this entry are the bottom four, so with just a little algebra on autopilot we get this, what are those coefficients in front of?

The powers of three one three three three one are very familiar, for example, the three in front of 3 to the power of 1 is equal to the three different trips from the circle 3 to the power of 1 to the orange and similarly with the other coefficients in the sum What this means is that those coefficients 1 3 3 1 are just the numbers in the fourth row of Pascal's triangle and that means that our sum equals an instance of the binomial formula one of those distant high school and college things. encapsulation of Pascal's triangle remember now and setting a equals 1 and b equals 3 you get our power of 4 springing from our nice powers of three and all of this easily generalizes to produce all the other powers of four on the left and so on It was like Karel Post demonstrated the

miracle

of humidity, a very nice and ingenious test, don't you think it made my day the first time I found it?

Well, that's all for today until next time.