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Middleton Tyas C of E Primary School sent some good submissions:

Seth wrote:


Having tried this activity a few times, I found the following method easiest:
     
First, I divided by 10. The answer given was a remainder of 9 so I knew that the number ends in 9 i.e: 19,29,39,49,59,69,79,89,99.
     
Next, I worked out which numbers go into the numbers above i.e: 3(39,99), 7(49) and 9(99). I found using the larger numbers helped so I divided by 9, this gave me a remainder of 0. The answer in this instance has to be 99!

Catherine wrote:

I divided the number by two and then by five at the beginning because it would give me the digit that the number ended in. If the remainder was three when divided by five it could end in three or eight. (10 + 3 = 13 and 5 + 3 = 8) I could work out which one by looking at whether it was an odd or even number which I knew by dividing by two.

I looked back and realised I could find out what digit the number ended in just by dividing by ten which made sense when you see that 2 x 5 = 10. Even though I could work out the last digit with one division there were still ten possible answers that it could be. I would have to make at least one more division to only have one answer. I decided to divide by nine as it had a big variation of remainders. For example, when you divide by three the remainders can be zero, one or two. If you divide by nine the remainders could be zero, one, two, three, four, five, six, seven or eight.  I tried out my method of dividing by ten and then nine a few times and found that there was only one possible answer it could be. I also scored maximum points in the game. I think this must be the easiest way to work out the number and essentially win The Remainders Game. 


Lehan and Cody from Sheldon College in Australia wrote and sent in a diagram.

Hi, me and Cody had found a solution and it has a 9/10 chance of working each round. The solution makes you figure out the randomly generated number in just two questions.

The method is that first time you divide by 10 and the second time you devide by 9.  Here is an example:  number: ???

My number divided by 10 has an remainder of 6
My number divided by 9 has an remainder of 5

In this case you know that the number is going to end in a 6 as divided by 10 it has a remainder of 6.
The next step is that you add the remainder amount when divided by 9 onto the 9 (in this case 9 + 5 = 14) and see if it ends in 6. If it doesn't, just go 2 x 9 + the remainder (in this case 2 x 9 + 5 = 23). If it still doesn't work then just go 3 x 9 + the remainder and so on until you find a number that ends in a 6 (note the 6 is just an example here as it is the remainder of the number divided by 10).  In this case it would be 9 x 9 + 5 = 86 so 86 will be the answer.

There are only two types of questions that this method would not work on.

Question 1:
My number divided by 10 has an remainder of 2
My number divided by 9 has an remainder of 2

In this case, the number ends in a 2 because the number devided by 10 gives you a remainder of 2 however the number divided by 9 also gives you a remainder of 2. In this case there are 2 possible choices: 2 or 92. The number 2 doesn't go into 9 or 10 so it stays as it is while 92 divided by 10 gives a remainder of 2 and 92 divided by 9 gives a remainder of 2. There are no other possible choices. Right now you have two choices of either guessing between the 2 as you have a 1/2 chance or just ask another question. If you face this question it will be the remainder of the number divided by 10 or 90 + the remainder of the number devided by 10.

Question 2:
My number divided by 10 has an remainder of 9
My number divided by 9 has an remainder of 0

In this case, the number ends in a 9 because the number divided by 10 gives you a remainder of 9 however the number can go into 9 without remainders so again there are two posssible answers: 9 or 99. Both of these numbers go into 10 with an remainder of 9 and both of these numbers go into 9 without remainders. So in this question, again you ether make a guess or ask another question.

Other then these two types of tquestions this method works perfectly. Here is a  photo with the circles labled 1 - 100 just show better how it works:


 

Thank you for these wonderful responses.  Well done!