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(May 2014) Now some work from long-time contributor Maribeth Dann's class. Think about it carefully before reading their classmates' corrections, which appear below it. (Look at the first solution too, at the bottom, from a different school.)

We enjoyed your alien puzzles, they were big fun for us. They were very challenging to solve.

To solve the first question, how to tell if a number in Diphland was even, one of the group members had the brilliant idea of the head and the tail both being even, for example if the number was 6:2 then the solution would be the number four. But let's say the number was 9:2 then the solution would be five. We also tried some two-digit numbers with this theory, likewise 28:12 therefore being 16 as the answer. After experimenting on a few numbers we finally concluded that if the head and tail are both even, then the number is an even number. But even though subtracting two odd numbers gets you an even number in our system we could not do that because the tail could not be odd since the tail could only use the even numbers of 2, 4, 6, and 8.

The next question we encountered asked us to come to a conclusion of the easiest way to determine if a Diphland number is a multiple of three. Our group's first attempt was to talk our way through what did and didn't work, meaning we discussed all of the possible number sequences to decide if they were possible. We then came to the conclusion that if both the head and the tail were divisible by three, that would make the solution divisible by three. For example, one number sequence that we tried out to find the solution were 9:6. Both numbers are divisible by three and when you solve it, the result is three itself. We continued the process by testing the sequence of 36:24 to conclude that it works also. As one last check we tested 21:9. It too worked, and we concluded that when both the head and tail are divisible by three, the result is too.

The next question we ran into was how can you tell if a number is divisible by ten in Diphland. Solving this question was a real challenge for our group. To solve this one, we were in our group thinking of ideas all of which did not work until one group member thought of the idea that if they both ended in the same number then they would be divisible by 10, for example, 18:8 would add up to 10 so it would be divisible by 10.

The last question we encountered asked if it was possible to represent 6 in the Diphland number system. To answer this question, our approach was to, again, test out number sequences. Our first attempt was 7:1, but the digit 7 is never used in Diphland's number system. We then tried 8:2, but 8 cannot be used in the head. We determined that you would never be able to make 6.


The 7th Grade Enrichment Students at Hermann Middle School

eeps responds...

There are many intriguing things about this response. Here's one: look at their idea about being divisible by ten. Makes sense, right? But their example has a problem. What is it? You can use that problem to make the answer (which seems correct) more specific.

Then, look at the example in doing multiples of three. Their last check was 21:9. But that's not a number in Diphland!! (Why not?) But, but ... does their idea work anyway? (By the way, there is a way to refine their answer to this question and make it slightly simpler. We like simpler answers when we can get them. :)

Finally, see how they explain their ideas? Writing about math is HARD. I think they do a good job of explaining what they're thinking. How do they do that? What makes the explanations clear? What could they do to make them clearer?

The very next day, the teacher, Maribeth Dann, wrote, "I have another group that was more than happy to fix the mistakes in their classmates response!"

Here it is:

Dear Tim,

We are students, also from Hermann Middle School, who are going to correct our classmates mistakes. Our classmates first mistake was the question about how you can tell if a number in Diphland is divisible by 10. The only way a number can be divisible by 10 is for the number to end in six. This is because of the numbers we were given. With three, six, and nine being in the head, and two, four, six, and eight being in the tails, the only numbers that could be 10 apart would be a number that ends in six.

The next mistake our classmates had was the question about the number being a multiple of three. In our base ten system, a number is divisible by three if the digits in the number add up to a multiple of three. For example, 27 is a multiple of three because 2+7=9, and nine is a multiple of three. In Diphland it is the same way, but only the tail has to be a multiple of three because the head will always be a multiple of three, because any combination of three, six, and nine will always give you a multiple of three. So, the easiest way to tell if a Diphland number is a multiple of three would be to tell if the tail is a multiple of three. Our classmate used 21:9 as an example, but that wouldn’t work because you can never use one, two cannot be on the heads side, and nine cannot be one the tails side.

Finally, our classmates did a good job of explaining what they were thinking by giving examples. They could have given better, more accurate answers, but they did give good examples.


The Smartest of the Smartest Seventh Graders at Hermann Middle School

Hooray! A FIRST solution to Diphland! Once again, from eighth-graders at Monroe City Middle School in Monroe City, MO:

Dear Tim,

We found a solution to Diphland. Here are the answers to your questions.

What's the easiest way to tell if a Diphland number is a multiple of three?

If the head minus the tail is a multiple of three.

In Diphland, how can you tell whether a number is divisible by 10? Do you think you can make all the numbers that are divisible by 10?

If the last number of both the head and tails is six. For example, 66:46= 20. All the numbers devisible by 10 are 36:26, 66:46, 66:26, 96:66, 96:46, 96:26, 96:86.

Minh thinks it's impossible to represent the number 6 in Diphland. Can you make a convincing argument one way or the other?

No, you can not make 6 because you can't use odd numbers on the tail.

What is the easiest way to tell if a Diphland number is even?

The easiest way to tell if a number is even is to see if the last digits in the head and tail are even. If they are the number is even.

Your good old friends from Monroe City,

Jennifer, Joslynn, Matthew

teacher: Mrs. Barbara Carson, Monroe City Middle School

Thank you for your solution! That crowd in Monroe city sure does some good work. I invite folks to maybe clarify the reasoning here. For example, you say "you can't make 6 because you can't use odd numbers in the tail." That may be true (or it may not) but why does not using odd numbers in the tail mean you can't make a six?

Also: looking at the multiple of three answer, the head minus the tail is the value of the number, so indeed, you can figure out if it's a multiple of three by computing the value. But I'm wondering if there's an easier trick. For example, in our system, you can add up the digits, and if that's a multiple of three, the number is a multiple of three. Is there any such trick in Diphland?

You know, you can answer these questions even if you don't live in Monroe County, Missouri! Show me! Just write to !

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