Onmatu- General Math Learning Article

Amazing simple Math problems


          We will start with a simple problem: we draw a circle and we split that circle in 10 equal regions like in the image below:
 
                                                               Image result for circle divided 10 equal parts     

          Then we make the horizontal diameter more thick so that is more clear which one is the upper half and which one is the lower half like in the image below: 
   
                                                               

          Now we have the digits from 0 to 9 so in total 10 digits. We can put each digit in each section of the circle like this:  

                                                               

          This is just an example of how digits are arranged in the sections of the circle and the condition is that the digits can't be repeated so the rule is: each section of the circle has a different digit from those 10. Your task is to answer the following questions:

          a) How can you arrange the digits so that the sum of the digits in the upper half is equal with the sum of the digits in the lower half?
          b) How can you arrange the digits so that the product of the digits in the upper half is equal with the product of the digits in the lower half?
          c) How can you arrange the digits so that if you divide the bigger digit to the smaller digit for every pair of two opposite sides you get a whole number? (for example you choose a pair of two opposite sides, let's say 1 and 6 ; if you divide the bigger digit to the smaller digit you get: 6:1=1 which is a whole number)  
          d) How can you arrange the digits so that if you add the bigger digit with the smaller digit for every pair of two opposite sides you get always the same sum?
          e) How can you arrange the digits so that if you subtract from the bigger digit the smaller digit for every pair of two opposite sides you get the same difference?
          f) How can you arrange the digits so that if you multiply the bigger digit to the smaller digit for every pair of two opposite sides you get always a product equal or bigger with 9? (for example you choose a pair of two opposite sides, let's say 3 and 4 ; if you multiply the bigger digit to the smaller digit you get: 3*4=12 which is bigger than 9) ? What about getting every time a positive product?

         Thank you for your time
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