Mathemagician Video

[Valentin Albillo]

(07-24-2021 01:29 AM)jlw Wrote:  Then again, in at least one of my HP calculator manuals, is the story of Truman Henry Safford, a 10-year-old child prodigy asked to square 365,365,365,365,365,365. A parting shot claimed that no calculator ever made, even theirs, could do that.

Actually, it's much easier than it seems, I can do it as well. The ease comes from the fact that this is a especial-form number, namely

365 365 365 365 365 365 = 365 x 1 001 001 001 001 001

To square this number you must square both factors and then find their product. The square of the second factor is trivial, namely

1 002 003 004 005 006 005 004 003 002 001

while the square of 365 is also trivial to do mentally as it ends in 5. Namely, its square is

365^2 = [36 x 37][25] = [36^2 + 36][25] = [1296 + 36][25] = [1332][25] = 133 225

which is trivial to compute very quickly assuming you know the squares of integers up to 50 from memory, which all mental calculators do, myself included. Else, computing the square of 2-digit numbers such as 36 is pretty trivial, too: 36^2 = 36 x 6 x 6 = 216 x 6 = 1296

Now, all you need to do is compute 133 225 times 2,3,4,5 and 6, which againt it's very easy to do due to its especial form once more:

133 225 x 2 = 266 450, 133 225 x 3 = 399 675, etc

Now you keep in mind these six six-digit numbers and add them up correctly shifted by 3 positions each time and minding the carries and there you are:

365365365365365365^2 = 133 491 850 208 566 925 016 658 299 941 583 225

V.