Diphland
(page 33)

Hooray! A 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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