Digital Puzzles 35very easily found. For example, the squares of 1, 11, Ill, and 1111 are respec-
tively 1, 121, 12321, and 1234321, all palindromes, and the rule applies for any
number of l's provided the number does not contain more than nine. But there
are other cases that we may call irregular, such as the square of 264 = 69696
and the square of 2285 = 5221225.
Now, all the examples I have given contain an odd number of digits. Can
the reader find a case where the square palindrome contains an even number
of figures?
- FACTORIZING
What are the factors (the numbers that will divide it without any remainder)
of this number-l 0 0 0 0 0 0 0 0 0 0 0 I? This is easily done if you happen to
know something about numbers of this peculiar form. In fact, it is just as easy
for me to give two factors if you insert, say, one hundred and one noughts,
instead of eleven, between the two ones.
There is a curious, easy, and beautiful rule for these cases. Can you find it?
- FIND THE FACTORS
Find two whole numbers with the smallest possible difference between
them which, when multiplied together, will produce 1234567890.- DIVIDING BY ELEVEN
If the nine digits are written at haphazard in any order, for example,
4 1 2 5 3 97 6 8, what are the chances that the number that happens to be
produced will be divisible by 11 without remainder? The number I have
written at random is not, I see, so divisible, but if I had happened to make the
1 and the 8 change places it would be.
- DIVIDING BY 37
I want to know whether the number 49,129,308,213 is exactly divisible
by 37, or, if not, what is the remainder when so divided. How may I do this
quite easily without any process of actual division whatever? It can be done
by inspection in a few seconds-if you know how.