3-Digit Numbers Whose Digits Have a Sum of 6 or ___ (Feb. 7-13)

Recently, I was tutoring an 11th grade student in SAT math. She was unable to answer a question not because she couldn't add; she had no strategy; she did not look for patterns; she failed to use a systematic approach to problem solving. 

Here's the question: how many three-digit numbers have digits that sum to the number 6?

Here was her approach: 222, 402, 114, 510, 330, and so forth. She found 16 of the 21 solutions. Not only did she get the problem wrong and lost a quarter point, but she lost over 5 precious minutes when the problem should have taken less than a minute.

A Systematic Approach: When I asked her to think of creating a process by which she would start with the largest 3-digit solution (600), then move to the next group of hundreds (501 and 510), and then the next group (402, 420, and 411). She immediately realized that this pattern of 1 + 2 + 3 solutions would continue with 4 + 5 + 6 = 21. She did not have to even find the final 15 solutions to solve the question.

My objective: I have no interest in how my Mathletes do on the SATs although secretly I know they will all score in the high 700s. My main objective is to teach students of all ages to solve problems with a systematic approach, a logic, a process; to be more efficient with their time, and finally, to see the beauty in patterns. See below.

Number

of Solutions

for each

Hundred Solution(s)

1 600

2 501 510

3 402 420 411

4 303 330 312 321

5 204 240 213 231 222

6 105 150 114 141 123 132

21 solutions

Mathlete Lesson: 

After having the Mathletes discover this process on their own, they were able to answer this question in just minutes. The fact that they will not take the SATs for 6-10 years is irrelevent.  Then I asked them to go further. start with a sum of 1. How many three-digit numbers have digits that sum to 1. Of course, they immediately knew the answer was one solution (100).

Then they tried a sum of two: 

200

101 110 (answer: three)

Then they tried a sum of three, four and so on. The attached pdf will give them worksheets on which to continue with this project.

The Weekly Challenge: The Mathletes should keep going until they find the sum with the most number of solutions. The last attached pdf provides solutions up to the sum with the largest number of solutions. incidentally, the highest three-digit number (999) has a sum of 27; the sum of 26 has only three solutions -- do you see the pattern?

Mathlete Dollars: Any Mathlete who finds all of the solutions from 1-27 will earn 100 Mathlete Dollars. Of course, there is no requirement to find all of the solutions. They should try their hardest and have fun. Work on this with your children.

Thursday's Kindergarten Class: Although we did not do the same challenge as described above, we did a similar exercise and I would like you to try this with your children this week. Roll a six sided dice and whatever number appears is the sum number. So when I rolled a three, I asked the children to find all of the three-digit numbers that had a sum of three. They found all six solutions (300, 201, 210, 102, 120, and 111). 

Of course, first we had to learn more about three digit numbers and why the numbers 003 and 030 were not solutions because they were only one and two digit numbers, respectively. It was fun just listening to them recite the numbers. They wanted to do more so we rolled again and landed on four. A few of them saw the pattern and started with the number 400, then 301, then 310, and so on until they found all ten solutions.

I left them with the challenge to find all 21 solutions that sum to 6. Before class ended, some of them had found up to eight solutions and counting. 

Work with your children and have fun this week.