Very often a problem situation seems so simple that
we plunge right in without first thinking about a strategy to use. This
impetuous beginning for the solution often leads to a less elegant solution
than one that results from a bit of forethought. Here are two examples of
simple problems that can be made even simpler by thinking before working on
them.
Find all pairs of prime numbers whose sum equals 999.
Many will begin by taking a list of prime numbers
and trying various pairs to see if they obtain 999 for a sum. This is obviously
very tedious as well as time consuming.
Let’s use some logical reasoning to solve this
problem. In order to obtain an odd sum for two numbers (prime or otherwise);
exactly one of the numbers must be even. Since there is only one even prime,
namely 2, there can be only one pair of primes whose sum is 999, and that pair
is 2 and 997. That, now, seems so simple.
A second problem where preplanning, or some orderly
thinking, makes sense is as follows:
A palindrome is a number that reads the same forward and backward, such as 747 or 1,991. How many palindromes are there between 1 and 1,000 inclusive?
The traditional approach to this problem would be to
attempt to write out all the numbers between 1 and 1,000 and then see which
ones are palindromes. However, this is a cumbersome and time-consuming task at
best, and one could easily omit some of them.
Let’s see if we can look for a pattern to solve the
problem in a more direct fashion.
Range
|
No. of palindromes
|
Total number
|
1–9
|
9
|
9
|
10 –99
|
9
|
18
|
100 –199
|
10
|
28
|
200 –299
|
10
|
38
|
300 –399
|
10
|
48
|
.
|
.
|
.
|
.
|
.
|
.
|
.
|
.
|
.
|
There is a pattern. There are exactly 10
palindromes in each group of 100 numbers (after 99). Thus, there will be 9 sets
of 10, or 90, plus the 18 from numbers 1 to 99, for a total of 108 palindromes
between 1 and 1,000.
Another solution to this problem would involve
organizing the data in a favorable way. Consider all the single-digit numbers
(self-palindromes), which number 9. The two-digit palindromes (two same digits)
also number 9. The three-digit palindromes have 9 possible “outside digits” and
10 possible “middle digits,” so there are 90 of these. In total, there are 108
palindromes between 1 and 1,000, inclusive.
Clever counting can often make work much easier.
The motto is: Think first, then begin a solution!
The triangular palindrome can also be expressed in an alternative arrangement as a fraction. Additionally, its denominator can be expressed as a perfect square. Read more here: http://www.glennwestmore.com.au/the-triangular-palindromes-secret/
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