Math challenge - For students only

Jeff Green | May 21, 2014

MATH CHALLENGE - FOR STUDENTS ONLY

ALL AGES

The Frontenac News has just finished a contest whose participants had to find the largest number with all its digits different and which is the sum of two numbers that involve at most four distinct digits. Now it is the turn of our young people to continue the investigation.

THE STORY SO FAR

In the equation 8887766555 + 88776655 = 8976543210, all the digits of the sum are different, and the two summands involve only the four digits 5, 6, 7, 8. This is not the largest possible sum. The largest number with all its digits different is 9876543210, and this can be written as the sum of two numbers involving only four different digits in 1712 ways, one of which is

9776766555 + 99776655 = 9876543210:

A different example provides a summand that has only two distinct digits:

6556555 + 9869986655 = 9876543210:

However, if we allow only three different digits in the summands, we can find several examples, such as

767666 + 97997766 = 98765432 and 816161186 + 818816866 = 1634978052:

NOW IT'S YOUR TURN

There are still outstanding questions. The Frontenac News will offer awards to students who write the best reports. Below are a number of questions that you might consider. You can focus on one or two of them, or give a more comprehensive survey. In all cases, the digits of the sum all have to be different.

(1) What are the largest sums that can be written where the two summands involve only one distinct digit? two distinct digits? three distinct digits?

(2) Suppose that each of the summands is required to have only two distinct digits, and together have at most two, three or four distinct digits? What now is the largest sum in each instance?

(3) In the various cases indicated above, determine that largest number of the digits the sum can have, and for this number of digits, we want to make the total number of digits (counting repetitions) in the three numbers as small as possible.

(4) Find examples of sums that are interesting in some way. For example, the digits might appear in each number in increasing, rather than decreasing order; some of the numbers could be palindromes (reads the same from left to right or from right to left); you could have summands with two distinct digits that alternate. The possibilities are left to your imagination.

(5) You might consider similar problems involving subtraction or multiplication of two numbers.

Here are the rules: There should be no reference to books or the internet, but a calculator or computer can be used to cut down on the tedium of searches and calculations. If you use a computer, give an informal description of how you set up the program. Solutions can be submitted by individuals or by groups; the name of each person contributing to the solution should be listed along with his/her age, grade and school. Teachers are welcome to collaborate with their students provided the students themselves make a significant contribution.

DEADLINE FOR ENTRIES - Friday. May 30.

There will be separate categories of awards for elementary students (up to approximately 11 years of age), senior elementary students (aged approximately 11-14) and secondary students.

Please send your solutions to the

Frontenac News, Box 229, Sharbot Lake ON, K0H 2P0

Tel: 613-279-3150; fax 613-279-3172

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