422 has three even digits. How many factors could it possibly have? Scroll down to see.

This Find the Factors puzzle has the same color scheme that puzzles 414 and 417 had, but this one also has a couple of numbers that aren’t colored at all. If 422 were colored, it would be yellow. I’ve had one very close guess posted in the comments for 417. Will anybody be able to read my mind and figure out what the coloring is all about? You can type your guess in the comments. Only an elementary education is required to figure it out!

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-09

- 422 is a composite number.
- Prime factorization: 422 = 2 x 211
- The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 422 has exactly 4 factors.
- Factors of 422: 1, 2, 211, 422
- Factor pairs: 422 = 1 x 422 or 2 x 211
- 422 has no square factors that allow its square root to be simplified. √422 ≈ 20.5426

**A Logical Approach to FIND THE FACTORS**: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row). ** Because this is a level three puzzle**, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

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Hmm, well the blue share 3 as a factor, the green shares 6 as a factor, and the yellow shares 2 as a factor. 2 times 3 equals 6, and yellow mixed with blue equals 6, I mean, makes green. 🙂

Exactly right! Thank you for your response!

So far, I think we have Greens are multiples of 6; Yellow are even multiples which don’t have a factor of 3; Uncoloured are prime; and Blue are odd multiples of 3. I think there may be a couple of loose ends – we don’t know for sure whether Blue numbers are simply the product of odd factors rather than 3 being essential (e.g. what about 35?), and I have to guess that 2 is Uncoloured – but I think / hope everything’s consistent with what I’ve said. That was fun!

I now see that I’ve left room for ambiguity! What you have written is just as legitimate as what I was originally thinking.

Here is how my original thinking process would have colored the numbers you mentioned: If there had been a 2 in the puzzle, it would have been yellow, and 3 would have been blue. All other prime numbers would not have been colored. 35 would not have been colored.

Thank you for noticing these numbers. If I do something like this again, I will try to anticipate other possibilities.

The colour problem was rather similar to challenges I’ve used. There’s a rather long article you can find at http://www.nrich.maths.org/7094 where I talk about the use of these games to model the processes of scientific discovery. The article will also direct you to several enjoyable interactivities on NRICH – “See The Light” and several others: Light the Lights, Light the Lights Again, Charlie’s Delightful Machine, Four Coloured Lights and A Little Light Thinking.

I really enjoyed your story about learning woodworking. I had no idea there was such a great story behind the chair in your photo!

As you said, the article is rather long, and I have not as yet completely read it through, but I will. Thank you for sharing it with me!