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Tome for a new keybiard

Today is the one hundred and eighth daily episode of my reports from an 8th decader's lockdown.

For a mathematician, 108 is a "good" number. Having three digits just locks into parts of the brain that tune into threes. And at a glance, it is a number that is divisible by three. Why, at a glance? Because if you add up the digits one, zero and eight, the answer is nine. Any number whose digits add up to a number that divides by three is itself divisible by three.

If after the first add, you have answer bigger than nine, add the digits together and keep doing that until you have a single digit. This is a digit sum.

If the final digit is a nine, then the original number will be divisible by three and by nine. If it's a six, then it's divisible by two and by three. And finally, if it's a three, then it is an odd number which is divisible by three.

I am far from sure, but my memory is trying to persuade me that I was taught this at school. I am certain about the rule of elevens. I was taught that. That rule means adding up each alternate digit in a number. And then add up the ones you didn't add in the first time. If these two answers are the same, or are a multiple of eleven apart, then the original number is divisible by eleven.

I have read in many books and seen in TV's Porridge, tales of fiction every one, that prisoners count the bricks in their cell in a desperate attempt to fight off boredom and mental deterioration. We do have a wall of stones in the house where I could do that. And our stone floors are marked in slabs. I have been incarcerated for 108 days.

A prisoner who should have my respect as a mathematician was Jakow Trachtenberg. What I have been writing about is what made him a person remembered today. He was a pacifist who was born in Odessa in 1888. Now that's another nice number. Looks good. Some will remember it as the foundation year that appears on the Celtic FC badge. For me, it is when the original Peterhead Prison opened. It's a prison museum; and well worth a visit. Not just to count the bricks in the walls.

Back to Jakow. As many Jews did, he fled from Russia (Odessa was then part of the Russian Empire) to Germany. As a critic of the nazis and a Jew, he had to flee again, this time to Austria. But when Hitler invaded in 1938, it was inevitable that he ended up in a concentration camp. I can't find the story of how he survived seven years in such a camp. In the end, he was due for execution but managed to escape.

He found an alternative to counting bricks on the wall to keep his sanity. He engaged in thinking about why people found arithmetic so hard. And developed the system now known as The Trachtenberg Speed System of Basic Mathematics.

I use the basics of the system. And have a view that one problem it might enable me to solve, by an algorithm rather than by brute computing force, is the finding of two very large prime numbers that have been multiplied together to make a larger one.

Why does that matter? It's not just an abstract mental exercise. The inability to do this is what underpins much of the security that protects messages on the Internet. I may have to stare a wall for some time to crack this one.

Just for the fun of it, I used Jakow's system of mental arithmetic to work out two to the power forty. Start with two and multiply two by two, getting four. Multiple four by four, getting sixteen. Multiply sixteen by sixteen to get 256. We're now at two to the power eight. Multiply by four to get to two to the power ten. We're at 1,024.

Multiply 1,024 by 1,024 to get 1,048,576. Let's pause for a moment. Back to digit sums. Add one, zero, four, eight, five, seven and six to get thirty-one. Add three and one to get four. Why do this? It's part of Jakow's checking process. Add one, zero, two and four to get seven. So multiply seven (like we did 1,024 times 1.024) to get 49. Add four and nine to get 13. Add one and three to get four. If that number is not the same as the digit sum for the big number, we did the multiplication wrong.

Have you lost the will to live yet? One more step. 1,048,576 times 1,048,576 gives 1,099,511,627,776. Do the digit sums again and lo, it works.

The bottom line about all this is that we all have to find our own way to manage the space in our thinking that something like lockdown can create. For 108 days, the amount of external stimulation I have received has been dramatically reduced. Forcing myself to sit down and write in the region of a thousand words each day has been absolute mental therapy.

Furthermore, it has assisted me in managing quite a few of the distractions which are always hanging around to reduce my efficiency and effectiveness. I do get invited into the kitchen to assist with administrating medicine to our older cat. I respond to a shout of "dishes". And I may roll ideas around in my brain during my use of the second-hand bath water the availability of which I am advised of by a call from ben the hoose.

But nothing serious has been started for more than a hundred of those days of incarceration until the writing's done. Some days, my brain works a bit slower. From time to time, the sheet of paper on my desk, which contains the jottings to provoke thinking and then writing on the following day, is blank. And sometimes the domestic intrudes.

I've rediscovered that old rule, don't start something new until you've finished what you are doing now. In part, that's been because at the beginning of the period; there were no formal intrusions arising from having to attend Parliamentary meetings. That gave me the space to re-establish some good habits.

Besides the approaching 130,000 words written so far, I have also learned a bit about letter frequency in the 5.285 different words I have used so far in my writing (thank you for the count Mr Grammarly.com).

As I look down at my keyboard, I've previously written that I am not a touch-typist, the letters "i", "o" and "t" have disappeared from the tops of their respective keys.

My mistyping rate is rising.

I had to buy a new pair of shoes after 400 miles.

Time for a new keyboard.

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