# Fay’s Nines Reflection

This past few weeks we have been doing the Fay’s Nines problem. Here are some unique solutions:

As you can see, there are 3 columns of numbers. The units column equals 19, the tens column equals 18 and the hundreds column equals 8. But for this to work, you have to carry (starting from the units column), then it all adds up to 999.

When I started playing with this problem I was interested to see how many unique solutions their were. I tried to find different strategies to work out different solutions, but I was having trouble finding unique solutions fast.

At the end of the first session (during the class discussion) Sienna and I realised that the units column had to add up to 19, the tens column had to add up to 18 and the hundreds column had to add up to 8. Here is our working out for the first session.

To break this problem into manageable parts, I got one unique solution and then turned it into another one by moving the numbers around. But with that strategy, I only got 2 solutions with similar numbers.

I kept doing trail and error for two sessions. Then I realised that if I got one solution, I could get a lot more. I didn’t realise it at the time, but I could get 36 more solutions. So, to start off with I kept the same numbers in the units column, and shuffled the numbers around those ones. I found 20 solutions, but I had a feeling there was more. After Mr Henderson had told us that there were 36 solutions for each 3 numbers that were in the units column, I had a light-bulb moment. Because there are 36 solutions for each combinations for each 3 numbers in the units column, and there are 5 units combinations (that equal 19), you multiply 36 by 5 and that equals 180, which is the correct answer.

This problem was challenging because I didn’t understand the carrying. After Mr Hopper helped my understand, I kept trying to find the answer. But little did I know that I had to find the answer to a smaller problem. To find the number of solutions for each 3 numbers in the units columns. This problem was very challenging and fun, and I hope to do something similar in the future.