When spring math circles were suspended in March, I started sharing some ideas for at home, independent math circles, posting them here until our summer camps started. From here on out I’ll post one idea each month where math circles don’t meet. Send any ideas you have to info@mathcirclesofchicago.org. Thanks!

#### Flips

Did you know that 1089 is a 9-flip? Let me explain.

1089 x 9 = 9801. See? Multiplying 1089 by 9 results in 9801, which is 1089 ‘flipped’.

- There are other 9-flips. Find one. Are there more?
- There are also 4-flips; numbers that you can multiply by 4 that result in ‘flipping’ the number you started with. Find a couple of 4-flips.
- Are there other kinds of flips besides 4-flips and 9-flips? Are there any 5-flips or 7-flips? If not, why not?
- Think about extreme cases. There are a lot of 1-flips. What is true about all of them? Can you have a 2 digit flip (like, say, a 19-flip)? Why or why not?
- What other questions about flips could you ask?

#### Tools

How could you approach the flip investigation?

- Guess and check. It gets us through life most of the time.
- Try a spreadsheet. This is sort of like guess and check on steroids. You can collect a lot of data fast and get a feel for what it takes to find/create a flip.
- Do some initial analysis. Take 9-flips. Multiplying by 9 results in a number much bigger than what you started with. That limits what numbers you might have in the first and last digits of the number you are playing with. With 4-flips, you have more options….
- Use Algebra if you know some. Suppose I have the number ABCD, where D is the digit in the 1s place, C is in the 10s, B in the 100s, and A is in the 1000s place, so that ABCD = 1000A + 100B + 10 C + D. How does that compare with DCBA? Of course, it’s better not to have too many variables, so use your analysis to narrow things down a bit. If you have a 9-flip, what number do you think will be in the left-most decimal place?

(Thanks to A. Gardiner’s *Discovering Mathematics: The Art of Investigation* for introducing me to Flips!)