Ivan crossed it all out and decided to begin right off with something very strong, in order to attract the reader’s attention at once, so he wrote that a cat had got on a tram-car, and then went back to the episode with the severed head. Michael Bulgakov,The Master and Margarita Let us our turn our attention to the emotional side of mathematics, more specifically, to the personal psychological experience of people working with mathematical algorithms and routines.
I wish to formulate here some of my observations and conjec- tures which may appear to be bizarre and out of tune from the usual discourse on mathematics. However, I tested some of them in a warm-up talk that I gave at a forum discussionWhere do math- ematicians come from? [452], part of a very peculiar conference, that of the World Federation of National Mathematics Competi- tions (WFNCM). It was held in July 2006 in Cambridge, England. On my way from Manchester to Cambridge, four hours by train, I had seen three people solving Sudoku. In one case, a lady of middle age shared a table with me and I had a chance to watch, in all detail and with a growing fascination, how she was solving an elementary level Sudoku puzzle. Her actions followed a certain rhythm: first she inspected the puzzle row by row and column by column until she located a critical cell (whose value had been already uniquely determined by the already known values in other cells), then, with obvious agitation, checked that was indeed the case, happily wrote the digit in, smiled with a childish satisfaction, relaxed for a few seconds, and, after a short pause, started the search again.
Next day, in my talk at the conference, I pointed out that, from a mathematical point of view, solving an elementary level Sudoku puzzle is nothing more than solving a triangle system of Boolean equations by back substitution, something very similar to what we do after a Gauss–Jordan elimination in a system of simultaneous linear equations. But has anyone ever seen people on a train solv- ing systems of linear equations from a newspaper?9
Why is Sudoku popular, when systems of linear equations are not? (Actually, I was slightly wrong: at the time of my talk, I was unaware of Kakuro which combines linear and Boolean equations. But one still has to see whether Kakuro beats Sudoku in popular- ity.)
1.6 Why are certain repetitive activities more pleasurable than others? 15
We will not understand the psychologi- cal and neurophysiological roots of an important aspect of mathematical prac- tice until we figure out why bubble wrap popping is such an addictive and pleas- ant activity.
Still, why is Sudoku popular? I be- lieve the answer is in a rhythm of repeated cycles of operations each of which engages our brains just up to a right and most pleasurable level of intensity. As a student, I experienced a soothing, relaxing effect of carrying out a recursive algorithm, like long division, or Euclid’s algorithm. Later, in my research work, I felt a similar
emotional impact of inductive arguments in finite group theory: you start with a minimal counterexample to the theorem, and then sim- plify it step by step, like removing layers from an onion, until you pin-point the core contradiction and destroy the counterexample. My teacher, Victor Danilovich Mazurov, expressed the principle of a “minimal counterexample” using a line from a Russian fairy tale:
The oldest brother hid behind the back of the younger one, the younger one hid behind the youngest one, and the youngest brother fell on his knees, raised his hands and pleaded for mercy.
Vladimir and Victor Mazurov ages 9 and 6 In mathematical education, especially at its earlier stages, one
of the teacher’s tasks is to give his/her students the opportunity to feel this soothing, comforting effect of a rhythmic repetitive activity. And here I come to the crucial point:
why do people love to pop bubble wrap?
I would not write this now if the audience of my talk at the WFNMC conference had not immediately agreed with, and ap- proved of, my comparison of execution of certain types of recur- sive algorithms with the bubble wrap popping. I should perhaps ex- plain that the audience included some of the best experts on math- ematical education in the world, especially on advanced and non- standard aspects of mathematics teaching. They definitely knew everything about “recreational mathematics”, puzzles, brainteasers and conundrums of every possible kind. Their support allows me to be quite confident in my comparison of Sudoku with bubble wrap popping. In any case, the lady on the train was doing her Sudoku in an immediately recognizable bubble wrap popping rhythm.
So, with the authority of the conference on my side, I dare to formulate my thesis:
We shall not understand the psychological and neurophysi- ological roots of an important aspect of mathematical prac- tice until we figure out why bubble wrap popping is such an addictive and pleasant activity. Why does it comfort and help to relax? Why is it soothing?
Actually, some years ago I formulated a rather embarrassing conjecture that the attraction to bubble wrap popping is genet-
ically determined. Bubble wrap triggers in humans archaic in- stincts linked to an ape-like behavior: grooming (and even more importantly, mutual grooming) and destruction of lice. In apes and monkeys, mutual grooming is an important part of social bonding, which explains its soothing, comforting, relaxing effect.
In my search on the web for a confirmation of my conjecture I have not managed to get further than numerous websites devoted to virtual bubble wrap popping. A search for the words “bubble wrap” is not the best way to find anything meaningful on the web: almost everything sold on the Internet is mailed in bubble wrap packaging. As the result, GOOGLEproduces 9,090,000 hits for “bub- ble wrap”. I offered the problem to my colleague Gregory Cherlin, who was more Internet savvy and carried out a successful search. Here are his principal findings:
• Gene HOXB8 controls normal grooming behavior. Disruption in mice leads to obsessive grooming behavior. Here is a summary of information from the National Institutes of Health website [443]:
This gene belongs to the homeobox family of genes. The homeobox genes encode a highly conserved family of tran- scription factors that play an important role in morpho- genesis in all multicellular organisms. Mammals pos- sess four similar homeobox gene clusters, HOXA, HOXB, HOXC and HOXD, which are located on different chromo- somes and consist of 9 to 11 genes arranged in tandem. This gene is one of several homeobox HOXB genes located in a cluster on chromosome 17. HOXB8 knockout mice ex- hibit an excessive pathologic grooming behavior, leading to hair removal and self-inflicted wounds at overgroomed sites. This behavior is similar to the behavior of humans suffering from the obsessive-compulsive spectrum disor- der trichotillomania.
• There is quite a range of grooming-related disorders in humans [449].
• Primates indeed do love to pop bubble wrap [440, p. 8].
Meanwhile, my own search for bubble wrap popping on Google Scholar led me to the book under the telling title Teens Together Grief Support Group Curriculum[425]. I have not seen the whole book, but, apparently, page 57 contains sufficiently revealing words:
Bubble Wrap: Give the teens a square of bubble wrap to pop for one of their breaks. They really get into the sound and action of popping the bubbles.
As I suspected, the soothing and comforting effect of bubble wrap popping is indeed well-known to practicing psychotherapists.