Reacreational Maths with Smartphone Calculator - 10 Games to Wow Your Friends

The problem with presenting an interesting party game is that your friends immediately ask for more.  Some will want to know how it was done.  For example, everybody gasps when the 4x4 magic square, after several arbitrary random choices, gives the sum as the correct birthday or marriage anniversary.  
Generally, you are asked - How do you get that?  Can you show some other games? etc.

I have decided to collect ten games that one can play on a four-function calculator that all smartphones have - and everybody carries one all the time!  After the initial groans when you ask them to take their smartphones out for some fun games, the interest builds up very quickly.  

Let us start:  Worked examples in red

Game 1:   On a piece of paper write 1089 and put the paper away. 
Now ask your audience to  

          1.  Choose a 3-digit number with different 1st and 3rd digits     357
          2.   Reverse the order of digits      753
          3.   Subtract the smaller from bigger  753 - 357 = 396  
          4.   Reverse the order of digits in the result.      693
          5.   Sum the numbers in steps 3 and 4              693 + 396 = 1089

The result is always 1089.  
Show the paper you had the prediction written on.

Note:  If the subtraction in step 3 is a 2-digit number then put a zero on the left side to make it a 3-digit number.

Game 2:   1.  Choose a 3-digit numer         425
                2.  Repeat the number to make it a 6-digit number.   425425
                3.  Divide the number by 7.    425425/7 = 60775
                4.  Add 715 to the result.         60775 + 715 = 61490
                5.  Divide by 11.               61490/11 = 5590
                6.  Subtract 65.                5590 - 65 = 5525
                7.  divide by 13.             5525/13 = 425

You have recovered your original number!!

Game 3:   1.  Choose a 4-digit number.            4798
                2.  Add digits.                4 + 7 + 9 + 8 = 28
                3.  Subtract sum of digits from the original no.  4798 - 28 = 4770
                4.  Cross one digit (except 0)  from the answer   4*70
                5.  Read out the remaining digits    4, 7, 0

In your head add the digits and subtract the sum from the next number that is divisible by 9.  
                      11; subtract 11 from 18 to get 7
                7.  The result is the deleted number.    

Game 4:   1.  Choose a 3 digit number (not all identical digits)     782
                2.  Form the biggest & smallest no. for these digits    872 and 278
                3.  Subtract smaller no. from the bigger number      872 - 278 = 596
                4.  Form the biggest and smallest no.   965  &  569
                5.  Subtract smaller no. from the bigger no.   965 - 569 =   396
                6.  Continue with steps 4 and 5 until you reach 495.
                7.  You has reached a limiting value that does not change anymore. 
You might need up to six steps to reach the limiting value.   

Game 5:   1.  Choose a 4 digit number (all digits not the same)     7821
                2.  Form the biggest & smallest no. for these digits    8721 and 1278
                3.  Subtract smaller no. from the bigger number  8721 - 1278 = 7443
                4.  Form the biggest and smallest no.   7443  &  3447
                5.  Subtract smaller no. from the bigger no.   7443 - 3447 =   3996
                6.  Continue with steps 4 and 5 until you reach 6174.
                7.  You has reached a limiting value that does not change anymore.
You might need up to 7 steps to reach the limiting value.

Game 6:   1.  Choose a 4-digit numer         7136
                2.  Repeat the number to make it an 8-digit number.   71367136
                3.  Add 6570 to this number.    71367136 + 6570 = 71373706
                4.  Divide this result by 73.         71373706/73 = 977722
                5.  Subtract 90 from this number.             977722 - 90 = 977630
                6.  divide by 137.       9777630/137 = 7136
You have recovered your original number!!

Game 7:  On a piece of paper write 18 and put the paper away. 
Now ask your audience to  

                1.  Choose a 2-digit number with different  digits   51
                2.   Reverse the order of digits      15
                3.   Subtract the smaller from bigger  51 - 15 = 36
                4.   Reverse the order of digits in the result.      63
                5.  Sum the numbers in steps 3 and 4              63 + 36 = 99
                6.  Sum the digits of the number obtained     9 + 9 = 18
The result is always 18.  
Show the paper you had the prediction written on.

Game 8:   1.  Choose a 2-digit numer         28
                2.  multiply the no. by 37       1036
                3.  Subtract 357       1036 - 357 = 679
                4.  Multiply result by 13   679 x 13 = 8827
                5.  Add 2587             8827 + 2587 = 11414
                6.  Multiply by 3                11414 x 3 = 34242
                7.  Add 6162             34242 + 6162 = 40404
                8.  Multiply by 7        40404 x 7 = 282828

You have recovered your original number repeated 3 times!!

Game 9:   1.  Choose a 2 digit number (both digits not the same)     38
                2.  Form the biggest & smallest no. for these digits    83 and 38
                3.  Subtract smaller no. from the bigger number  83 - 38 = 45
                4.  Form the biggest and smallest no.   45 and 54
                5.  Subtract smaller no. from the bigger no.   54 - 45 = 9  
                     If result is a single digit then put a zero on its left
                6.  Form the biggest and smallest no.   90 and 09
                7.  Subtract smaller no. from the bigger no.   90 - 09 = 81
                8.  Continue with steps 4 and 5.  You will find that you are 
                     repeating a string of numbers in the following order 
                     9, 81, 63, 27,45, 9, 81, 63, 27, 45, 9, .... 

Game 10:   The Amazing Number 1729 - Also known as the taxicab number.  
The story goes that Ramanujan, the mathematical prodigy, was in the hospital when his professor Hardy came to visit him.  When Ramanujan asked what the taxi number was - Hardy replied that it was an ordinary number 1729.
Ramanujan then said that 1729 is an amazing number because it has the following properties.

(a)   It is the smallest number that is equal to the sum of cubes of 
       two different sets of positive numbers  
             1729 = 1^3 + 12^3  = 9^3 + 10^3

(b)  If you add the digits of 1729, you obtain 19
      
      Now multiply 19 by its reverse i.e., 91 and you would obtain 19 x 91 = 1729
     Only three other numbers have this property.  
     They are   1, 81 and 1458

Try 1458 to see that it works.

Final Note:   I have algebraic proofs of the above games to show how they work.  If you are interested write to me at ektalks@yahoo.co.uk and I shall send you the proof you are interested in.

Enjoy!