To implement a programmatic solution (ie simulation) of a real life process- the mental math trick of computing the square of any number ending with 5.

The allure of using (and learning) a programming language is to be able to effectively use it to solve problems, which in and of themselves are simulations of some process we can do in “the real world”.

In this case, we will be writing a program which will implement the mental math techniques for computing the square of any two-digit number that ends with 5.

Mental Math constitutes an intersection of mental techniques and math- instead of utilizing a purely math-only solution, textual manipulations or simplifications in the computational process may take place enabling an individual to, once having learned the process, solve such problems in their head, and typically without the use of a calculating device.

The process in this case is one of numeric manipulation and simple (reduced) multiplication. To wit:

Squaring is essentially multiplying a number by itself-

• 5 squared (5^2) is 5*5 or 25
• 8 squared (8^2) is 8*8 or 64

While not outwardly a difficult procedure, the nature of multiplying multiple digit numbers in your head can quickly result in more steps (and more steps means more time, if doing things the traditional way).

Finding a shortcut through this process enables it to remain solidly within the realm of mental math, and makes for a good algorithm to practice implementing on the computer.

This particular trick relies on a subset of the squares: those ending with a 5 (a five in the ones place).

The implementational scope of this trick will be just values of one-, two-, and three-digits ending with 5:

• 5
• 15
• 25
• 35
• 45
• 55
• 65
• 75
• 85
• 95
• 105
• 115
• 125
• …
• 235
• 245
• …
• 485
• 495
• 505
• …
• 995

The trick here is two-fold. First, we separate the one's place 5 from the rest of the number (which can be accomplished in our mind's easily enough, but on the computer we must resort to some math).

We then take that isolated five and square it; we'll get 25. That is how our result will end (so bam! we now have our tens and ones place already solved)

Next, we take the remaining digits of the original value, and multiply it by its increment:

• 1's increment (1+1) is 2, so 1*2
• 2's increment (2+1) is 3, so 2*3
• 3's increment (3+1) is 4, so 3*4
• 4's increment (4+1) is 5, so 4*5
• …
• 9's increment (9+1) is 10, so 9*10

We take this result and append the 25 after it.

For example:

15 * 15 = 1*(1+1) 5*5
        = 1*2     5*5
        = 2       25
        = 225

and:

75 * 75 = 7*(7+1) 5*5
        = 7*8     5*5
        = 56      25
        = 5625

For a single digit number like 5, when you take the 5 away, what do you get? ZERO. Zero times anything is zero, so the result is 0 25, or 25 (ie this process still works).

For three digit numbers like 105, we have 10, and its increment is 11, so 10 x 11 = 110.

105 * 105 = 10*(10+1) 5*5
          = 10*11     5*5
          = 110       25
          = 11025

It is your task to write the program that will use the above method to compute the square of the input value ending with a 5 (you are to input the entire number, including the 5 at the end).

Your program should:

• prompt the user for the number (input)
  • input number as an unsigned short integer
• perform the task (process)
  • isolate one's digit mathematically, store in a variable (unsigned short int)
  • isolate remaining digits mathematically, store in another variable (unsigned short int)
  • perform the algorithm on the two pieces, storing their results in two separate variables (of type unsigned short int)
• display the final value (output)
  • display the beginning and ending parts together (but stored in separate variables)
  • display the resulting number to STDOUT
  • display any supporting text to STDERR

lab46:~/src/cprog/squares$ ./squares
Enter value: 75
75 x 75 = 5625
lab46:~/src/cprog/squares$ 

The execution of the program is short and simple- obtain the input, do the processing, produce the output, and then terminate.

Be sure to provide any commentary on your opus regarding realizations had and discoveries made during your pursuit of this project.

• Does this trick work (or can it be adapted to work) for three digit values ending with 5?
• How about 4 digits?

To successfully complete this project, the following criteria must be met:

• Code must compile cleanly (no warnings or errors)
• Executed program must display a total of 2 lines, one for input, one for output.
• Output must be correct, and match the form given in the sample output above.
• Code must be nicely and consistently indented (you may use the indent tool)
• Code must implement solution using the mental math technique described above
• Code must be commented
  • have a properly filled-out comment banner at the top
  • have at least 20% of your program consist of //-style descriptive comments
• Output Formatting (including spacing) of program must conform to the provided output (see above).
• Track/version the source code in a repository
• 25% late penalty per day after deadline
• Submit a copy of your source code to me using the submit tool.

To submit this program to me using the submit tool, run the following command at your lab46 prompt:

$ submit cprog mms0 squares.c
Submitting cprog project "squares":
    -> squares.c(OK)

SUCCESSFULLY SUBMITTED

You should get some sort of confirmation indicating successful submission if all went according to plan. If not, check for typos and or locational mismatches.