92.—­DIGITAL SQUARE NUMBERS.

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?

93.—­THE MYSTIC ELEVEN.

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.

94.—­THE DIGITAL CENTURY.

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 x) will thus count as two strokes, the sign of subtraction (-) as one stroke, the sign of division (/) as three, and so on.

95.—­THE FOUR SEVENS.

[Illustration]

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 99+9/9, but the four 7’s call for rather more ingenuity.  Can you discover the little trick?

96.—­THE DICE NUMBERS.

[Illustration]

I have a set of four dice, not marked with spots in the ordinary way, but with Arabic figures, as shown in the illustration.  Each die, of course, bears the numbers 1 to 6.  When put together they will form a good many, different numbers.  As represented they make the number 1246.  Now, if I make all the different four-figure numbers that are possible with these dice (never putting the same figure more than once in any number), what will they all add up to?  You are allowed to turn the 6 upside down, so as to represent a 9.  I do not ask, or expect, the reader to go to all the labour of writing out the full list of numbers and then adding them up.  Life is not long enough for such wasted energy.  Can you get at the answer in any other way?