Mathematical_Recreations-Kraitchik-2e

CHAPTER THREE
NUMERICAL PASTIMES
WRITE each of the numbers 1 and 100, using each of the ten
digits just once.
Solution: 1 = t&+ill, 100 = 50+49+t+H.
Write 31 using only the digit 3 five times.
Solution: 31 = 3 3 + 3 + t.
These problems are freakish, for their solution rests on no
general principle. They should really be classed as puzzles.
We have put them here only in order to give point to our protest
against their inclusion in an honest collection, so we have
not dignified them by giving them numbers.
FIGURE 2.
1. MISCELLANEOUS PROBLEMS
1. Explain the identity 1,111,1111-
(10 X 111,1112) = 1,111,111,111,111.
Answer: Write out the multiplication
of the first term as in Figure 2, but do
not add up the partial products. The
terms which are here enclosed clearly
represent the product 10 X 111,111'.
When these have been deleted, the remaining
digits represent precisely the
number 1,111,111,111,111.
2. Find four numbers such that the product of any two
of them is one less than a perfect square. (Diophantus)
Solution (by Boije af Gennas): Let a, b, c, and d be the
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