Fascinating numbers

Since I started learning to use the soroban, or Japanese abacus, I have also picked up some fascinating features of certain numbers.

Take 37 for example. I was given the finger exercise of 37 + 37 + 37 +37 +37 + ................ , quite an interesting challenge on the abacus, but imagine my surprise when one of the totals came up as 111, then later up comes 222 then 333, 444, 555, 666, 777, 888, 999. It turns out that this is because 111 is 3 x 37, 222 is 6 x 37 and so on.

I mentioned before that you can use the soroban to multiply. You actually do small number multiplication in your head and use the abacus to accumulate the results. This way you can multiply numbers of any size as long as your abacus is big enough to carry the result (mine goes up to 99,999,999,999,999,999 so I could multiply 26 302 648 by 467 984 567). All you need to know is your 2 through 9 times tables up to times 9 and since it is a highly competitive sport (against the clock), it helps if you know some shortcuts. Here are a few -

Any double number like 33, 66, 88 times 9. Multiply the number that is doubled by 9 and stick a 9 in between.

33 x 9 = 3 x 9 with a 9 stuck in the middle (first 27 then 297)

44 x 9 = 4 x 9 with a 9 stuck in the middle (first 36 then 396)

It works the other way round as well -

99 x 7 = 9 x 7 with a 9 stuck in the middle (first 63 then 693)

Any number times 5. Divide the number by 2, then if there is no remainder tack on a 0, if there is a remainder tack on a 5.

77 x 5 = 76 / 2 = 38 with a remainder, so the answer is 385

96 x 5 = 96 / 2 = 48 with no remainder, so the answer is 480

Any number between 11 and 19 multiplied by any single digit number. Multiply the second digit of the first number with the second number and tack the results together.

18  x 8 = 8 x 8 (64) tack the 8 onto the 6 = 144

17 x 3 = 7 x 3 (21) tack the 3 onto the 2 = 51

19 x 7 = 9 x 7 (63) tack the 7 onto the 6 = 133

and similarly, the other way round -

81 x 6 = 8 x 6 and tack on a 6 = 48 and a 6 = 486

61 x 9 = 6 x 9 and tack on a 9 = 54 and a 9 = 549

31 x 5 = 3 x 5 and a 5 = 155

Finally, talking of competition, Helen and I listened the other evening to a BBC Radio 4 programme covering the Soroban World Championships in Tokyo. One of the contests was to add 15 two digit numbers moving the beads of an imaginary soroban pictured in his head (this technique is called Anzan, pronounced "unzun"). The previous world champion broke his own record. This time he did it in 1,71 wait for it .... seconds!!! That meant he had to move imaginary beads at the rate of about 22 moves per second!

In the Level 10 Anzan exam which I am currently preparing for (that is one up from elementary), I have to add or subtract 10 sums each of 6 single digit numbers in 1 1/2 minutes. My best so far is 2 1/2 minutes that's about 1 1/2 seconds per move with three incorrect answers! It's a long way to Tokyo.

Fascinating stuff!

PS. Since writing this article a year ago, I've progressed and am now preparing for the 7th level exam (7th kyu). The anzan exam is now 10 sums of 4 double digit numbers, 10 double digit numbers multiplied by a single digit (like 24 x 7), 10 sums of 12 single digit numbers, 10 three digit numbers divided by a single digit (like 368/8), 10 sums of 4 digit numbers multiplied by a single digit (like 7243 x 8) and 10 four digit numbers divided by a single digit (like 9548/7). That's a total of 60 sums, all moving beads on an imagined abacus and the time allowed is 20 minutes. Incidentally, I don't aim to pass (80%), I practice for 100%. That way I consistently achieve above 95%, but it's still a long way to Tokyo.

 

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