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Even Sum Sudoku

Even Sum Sudoku

Today’s puzzle variant comes from the 2008 Sudoku World Competition Instruction booklet. I’ve renamed it Even Sum Sudoku for clarity. About a year ago, I published a cartoon called Odd Sudoku, where either Odd or Even contiguous Cells of at least size 2 were offered. This is not like that.

I’ve eliminated some starting numbers from the original puzzle and identified the cells in yellow as pairs with values summing to an even result.

One question that occurs to me is: what is the probability of having Even Sum Pairs for all the arrangements of this puzzle? Obviously, there are at least 8 Even Sum Pairs that have already been earmarked. From previous calculations (Domino Sudoku Cartoon), excluding the center cell, there are 40 pairs of contiguous cells in an arrangement (and there are 2**4 = 16 pair arrangements since:

  • In Row 1, columns 1 and 2 [A = Across] or the other starting in Column 1, Rows 1. 2 [D = Down]
  • In Row 2, columns 2, 3 [A] or Column 2, Rows 2, 3 [D]
  • In Row 3, columns 3, 4 [A] or Column 3, Rows 3, 4 [D]
  • In Row 4, columns 4, 5 [A] or Column 4, Rows 4, 5 [D]

Since each of these can be selected independently, there are 2*2*2*2 = 16 arrangements.

For any arrangement, how many are Even Pairs are there? It turns out, once you’ve solved the puzzle, you can count:

Across [A]   Down [D]
Odd: 8 Even: 8   Odd: 10 Even: 6
Odd: 9 Even: 3   Odd: 7 Even: 5
Odd: 3 Even: 5   Odd: 5 Even: 3
Odd: 3 Even: 1   Odd: 3 Even: 1
Totals:   Odd: 23 Even: 17   Odd: 25 Even: 15
Arrangement No. Odd Even
AAAA 1 23 17
DDDD 2 25 15
ADAA 3 21 19
AADA 4 25 15
AAAD 5 23 17
DAAA 6 25 15
ADDA 7 23 17
ADAD 8 21 19
AADD 9 25 15
DDAA 10 23 17
DADA 11 27 13
DAAD 12 25 15
ADDD 13 23 17
DADD 14 27 13
DDAD 15 23 17
DDDA 16 25 15
Frequency Odd   Even
2 21   19
6 23   17
6 25   15
2 27   13

P(Even = 19) = .125
P(Even = 17) = .375
P(Even = 15) = .375
P(Even = 13) = .125

A nice discrete, symmetric binomial distribution! Enjoy getting even.

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