38 Arithmetic & Algebraic Problems
- DIGITAL PROGRESSION
"If you arrange the nine digits," said Professor Rackbrane, "in three num-
bers thus, 147, 258, 369, they have a common difference of III, and are,
therefore, in arithmetical progression."
Can you find four ways of rearranging the nine digits so that in each case
the three numbers shall have a common difference and the middle number be
in every case the same?
- FORMING WHOLE NUMBERS
Can the reader give the sum of all the whole numbers that can be formed
with the four figures I, 2, 3, 4? That is, the addition of all such numbers as
1,234,1,423,4,312, etc. You can, of course, write them all out and make the
addition, but the interest lies in finding a very simple rule for the sum of all
the numbers that can be made with four different digits selected in every
possible way, but zero excluded.- SUMMING THE DIGITS
Professor Rackbrane wants to know what is the sum of all the numbers that
can be formed with the complete nine digits (0 excluded), using each digit
once, and once only, in every number?- SQUARING THE DIGITS
Take nine counters numbered I to 9, and place them in a row as shown. It
is required in as few exchanges of pairs as possible to convert this into a square
number. As an example in six pairs we give the following: 78 (exchanging 7
and 8), 8 4, 4 6, 6 9, 9 3, 3 2, which gives us the number 139,854,276, which is
the square of 11,826. But it can be done in much fewer moves.