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 reading a lecture 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. In his view we are literally swimming in a sea of these accumulated corpses — certainly by the time we’ve reached middle age but often well before that — and this makes it difficult to even notice the fact that we have the possibility of doing active thinking, to perceive new ‘living’ thoughts.
You can form an idea of what he is talking about by conceiving of our thinking as something closer to a sensory capacity, like seeing or hearing. We all know the contrast between looking (active) and seeing (passive) on the one hand, or between listening (active) and hearing (passive) on the other. Imagine one’s thinking as a sense organ which detects passing thoughts within the sphere of our awareness. This is not a spatial sphere as is the case with our field of vision, but the analogy still holds. In the same way that we choose and select what we see in rare moments of concentration, so can we control and orchestrate the permitted content of our thinking. This is a difficult discipline, and it is far more comfortable to simply relax and experience the ever unfurling horizon of thought pictures, like watching a movie with popcorn. But when a definite surge of will is exerted into our capacity for thinking, we can observe better where thoughts in our theater of consciousness arise from, and we can filter out the irrelevancies. We can ‘hunt’ for specific thoughts. Here, we can notice intuitions — thoughts which appear to arrive as ‘gifts’ unbidden from out of nowhere, which we more often than not discard and simply move on to the next scene in the movie. Making oneself receptive to these intuitions, and actively practicing taking note of them, and investigating where they come from and why — all this forms a modern spiritual practice which is appropriate for our cultural and intellectual milieu. (It requires little imagination to notice how the proliferation of the web as a mediator for a vast percentage of the thoughts which come to inhabit our individual clouds of clutter will act as a force against taking control of our thinking capacities in lieu of even cheaper, more abundant popcorn.)
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.