312 and Virgács

Today might not be the 312th day of the year, but here’s a fun fact about the number 312:

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  • 312 is a composite number.
  • Prime factorization: 312 = 2 x 2 x 2 x 3 x 13, which can be written 312 = (2^3) x 3 x 13
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 312 has exactly 16 factors.
  • Factors of 312: 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, 312
  • Factor pairs: 312 = 1 x 312, 2 x 156, 3 x 104, 4 x 78, 6 x 52, 8 x 39, 12 x 26, or 13 x 24
  • Taking the factor pair with the largest square number factor, we get √312 = (√4)(√78) = 2√78 ≈ 17.664

The night of December 5th, Mikulás, or St. Nickolas, will visit homes in Hungary, some neighboring countries, and even a few houses in the United States.  Children who have been good will awaken to find their shoes filled with little treats such as candy, fruit or nuts. Since all children are occasionally a little bit naughty, they will also find virgács, a small collection of twigs that have been spray-painted gold and decoratively bound together.  Virgács are not sold in the United States, so St Nickolas will be making some himself using the bristles from a natural broom.  The finished product should look like this:

This puzzle is meant to be reminiscent of the virgács my children’s ancestors probably received each year on St Nickolas Day.
2014-48 Level 3
Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-01
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.
2014-48 Level 3 Factors
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