Amusements in Mathematics HTML version

Here are the nine digits so arranged that they form four square numbers: 9, 81, 324, 576.
Now, can you put them all together so as to form a single square number—(I) the
smallest possible, and (II) the largest possible?
Can you find the largest possible number containing any nine of the ten digits (calling
nought a digit) that can be divided by 11 without a remainder? Can you also find the
smallest possible number produced in the same way that is divisible by 11? Here is an
example, where the digit 5 has been omitted: 896743012. This number contains nine of
the digits and is divisible by 11, but it is neither the largest nor the smallest number that
will work.
1 2 3 4 5 6 7 8 9 = 100.
It is required to place arithmetical signs between the nine figures so that they shall equal
100. Of course, you must not alter the present numerical arrangement of the figures. Can
you give a correct solution that employs (1) the fewest possible signs, and (2) the fewest
possible separate strokes or dots of the pen? That is, it is necessary to use as few signs as
possible, and those signs should be of the simplest form. The signs of addition and
multiplication (+ and ×) will thus count as two strokes, the sign of subtraction (-) as one
stroke, the sign of division (÷) as three, and so on.
In the illustration Professor Rackbrane is seen demonstrating one of the little posers with
which he is accustomed to entertain his class. He believes that by taking his pupils off the
beaten tracks he is the better able to secure their attention, and to induce original and
ingenious methods of thought. He has, it will be seen, just shown how four 5's may be
written with simple arithmetical signs so as to represent 100. Every juvenile reader will
see at a glance that his example is quite correct. Now, what he wants you to do is this:
Arrange four 7's (neither more nor less) with arithmetical signs so that they shall
represent 100. If he had said we were to use four 9's we might at once have written 999/9,
but the four 7's call for rather more ingenuity. Can you discover the little trick?