Advancing the calendar trick

I've written twice (first and second) about classroom experiences with a simple calendar trick that I originally got from Calendar Puzzles via Denise Gaskin's monthly newsletter. As happens so many times with these things, ideas from the kids make these activities into deeper and more interesting than I could have imagined on my own.

Tricked
On Tuesday, some of the first graders gave me their sums: 168 and 198. I immediately knew something was up. In the original calendar game, the square with the largest possible sum is the 23-24-30-31 square:

23	24
30	31

This has a sum of 108. I asked the students if they were sure of 168 and 198. They giggled, then the teacher smiled and told me she had checked it. What was going on?

My homework
I didn't have any immediate ideas, so I promised the kids that I would work on their puzzles. I told the kids that it was great to get my homework from them this time!

See through paper
One clue was that we were using a special calendar today and the paper was slightly see-through. This gave me an idea that the kids had turned the paper over and were seeing the numbers through the page with digits reversed. At first, I thought they were transforming 2s to 5s and vice versa, but was able to find 168 just by reversing digits.

How many carries?
For the original game, the crucial insight is simply that there are 7 days in a week and the calendar is organized into weeks. That means there is a simple relationship between each of the numbers in our 2x2 squares. Add a bit of simple algebra and you have an easy formula relating the upper left square of your 2x2 matrix to the sum (or, if you want to be fancy, a different formula relating whichever square you want to the sum).

For the reversed game, though, it isn't quite so easy. The relationship between the numbers can take one of several forms and is rather messy. I did manage to get 168. Can you?

But, I still couldn't get 198.

Two little helpers to the rescue
Last night, I "cheated" and asked for help. As J1 and J2 got ready to sleep, I asked what they thought their friend might have done to get 198. Their ideas from brainstorming:

• maybe the friend made an addition error
• maybe the friend also transformed 2s to 5s when reversing the paper
• maybe he summed a 3x3 square instead (which quickly gave rise to 4x4 and 5x5)

3x3 square? Interesting! Work through the algebra again and you can quickly see that there is (always!) a 3x3 square whose contents sum to 198.

Some further exploration, for you
More fun follow-on questions:

1. If the kids are allowed a choice of 2x2 or 3x3 section, but they still only tell you the sum and not the size of their square, can you still figure out which days they chose? Are there any conditions you might put on which month is chosen that allow you certainty in finding the square?
2. What if you allow 4x4, too?
3. Why stop at 4x4? What size squares are possible on a 1 month calendar?
4. If you make a year calendar instead, what sums are possible? If you are given the size and sum of a square, how close can you get to finding the source? In other words, how many squares have the same sums?

If you have other ideas, please let me know in the comments!