Addendum to Truth-See Demeanor

When I was writing this story I often was trying to put myself back into the mind of a little boy, like Icker. One day I especially was working on trying to get as clear as possible about the means and moment when a mathematical intuition becomes available, becomes suddenly present as a concept. I had been inspired by the following quote from Rudolf Steiner, who had been discussing the difference between what he termed ‘pure thinking’ and the more usual passive surveying of thoughts which happens during our conscious hours. Steiner viewed these latter objects of our contemplation as mere debris left over from previous thinking activity, either ours or another’s. He referred to them as thought corpses. From a 1920 lecture:

” “

I asked myself what other avenues could have opened as directions for a solution to Icker’s problem, as an answer to his intuition about a faster way. Each such route towards solving the problem, if it is a true one, reveals a different aspect of reality about the underlying thing being considered, in this case the sequence of numbers from 1 to 100. To keep looking is to invite seeing additional perspectives about some reality. But this is exactly what constitutes the reality, the spiritual essence, of a perceived thing. It’s many-faceted and active being.

The following avenue announced itself to me. Notice that the sum of the first ten numbers is fifty-five.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

How does this relate to the overall sum, the sum of all one hundred numbers?

If we take the next ten numbers, the so-called teens, we see a repeated pattern. 11,12,13,14,15,16,17,18,19,20. We see the same ten integers which we just previously summed up to 55, namely the numbers from 1 through 10. But in this case, each of these numbers is preceeded by the digit ‘1’ which signifies ten. To say it differently, the consecutive integers between 11 and 20 can be looked at as follows:

10+1 + 10+2 + 10+3 + 10+4 + 10+5 + 10+6 + 10+7 + 10+8 + 10+9 + 10+10

We’ve just already summed the single digits (blue part) and came up with 55. The rest (red part) amounts to 10*10 or obviously 100. So we have the result, with very little extra computational effort, rather insight, that the numbers between 11 and 20 tally up to 55+100 or 155.

Doing conceptual math is partly about seeing patterns, and we see one emerging here. The next ten integers, the so-called twenties, reduce as follows:

20+1 + 20+2 + 20+3 + 20+4 + 20+5 + 20+6 + 20+7 + 20+8 + 20+9 + 20+10

Or, 55 + (20*10) = 255.

Generalizing all the way up to 100 looks like this:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
10+1 + 10+2 + 10+3 + 10+4 + 10+5 + 10+6 + 10+7 + 10+8 + 10+9 + 10+10 = 100 + 55 = 155
20+1 + 20+2 + 20+3 + 20+4 + 20+5 + 20+6 + 20+7 + 20+8 + 20+9 + 20+10 = 200 + 55 = 255
30+1 + 30+2 + 30+3 + 30+4 + 30+5 + 30+6 + 30+7 + 30+8 + 30+9 + 30+10 = 300 + 55 = 355
40+1 + 40+2 + 40+3 + 40+4 + 40+5 + 40+6 + 40+7 + 40+8 + 40+9 + 40+10 = 400 + 55 = 455
50+1 + 50+2 + 50+3 + 50+4 + 50+5 + 50+6 + 50+7 + 50+8 + 50+9 + 50+10 = 500 + 55 = 555
60+1 + 60+2 + 60+3 + 60+4 + 60+5 + 60+6 + 60+7 + 60+8 + 60+9 + 60+10 = 600 + 55 = 655
70+1 + 70+2 + 70+3 + 70+4 + 70+5 + 70+6 + 70+7 + 70+8 + 70+9 + 70+10 = 700 + 55 = 755
80+1 + 80+2 + 80+3 + 80+4 + 80+5 + 80+6 + 80+7 + 80+8 + 80+9 + 80+10 = 800 + 55 = 855
90+1 + 90+2 + 90+3 + 90+4 + 90+5 + 90+6 + 10+7 + 90+8 + 90+9 + 90+10 = 900 + 55 = 955

But before adding up all these subtotals, it is advisable to once again step back.
Taking stock, we now have a total of ten ’55’s plus all the hundreds to sum up:

100+200+300+400+500+600+700+800+900 = 4500 + (10*55) = 4500 + 550 = 5050. Tada.

(I can tell you for a fact that Icker would have noticed that a further shortcut exists in tallying up the hundreds, because we have already added the digits from 1 to 10 and gotten 55. This is the same sequence except they are in hundreds, plus the fact that we’ve only got nine of them instead of ten. 55-10=45. 45*100=4500.)

This avenue is slightly slower than Icker’s solution but still much quicker than adding up all one hundred integers by hand. But the interesting thing is not the velocity. Again, it is that a new perspective about the texture of the numbers between 1 and 100 (and about numbers in general) has been revealed.

_______RS