Sebunzu

Newspaper puzzle pages have, in recent years, witnessed an explosion in grid-based number games. Since The Times featured their first Sudoku in 2004, countless spin-offs have made their appearance as publications compete for our coffee break procrastination time.

A Puzzle page from an edition of i, a UK newspaper,

Newspapers generally outsource the curation of their puzzles. Those in the illustration above are the work of Clarity Media whose clients also include three other major British newspapers and two publishers involved in creating educational materials.

Seeing as though it's half term, I thought I'd have a go at inventing my own newspaper-style puzzle rather than tear my hair out trying to complete someone else's. All that is needed is a repeatedly alterable puzzle-meme, an invariant set of instructions, and a catchy Japanese name.

After at least five minutes scribbling, I decided my puzzle would involve presenting the reader with a square grid containing a selection of integers between 1 and 5 and asking them to divide it into polyominoes each containing a total of seven.

For example, if I were to offer the following grid

You could simply divide it up as follows

As it happens, there is more than one solution here. We could swap the colours of the two 3s. Uniqueness is important, but a greater priority is to make it more entertaining. The puzzle is very easy at present and the natural way to complicate it is to enlarge the grid.

How about this?

This is marginally more engaging, but it is still possible to construct different correct solutions. These two, for example:

A problem with easily determining whether a solution is unique lies in the great flexibility with which we may partition the grid. Evaluating the number of possibilities is, I suspect, a seriously ugly problem and I can't find any literature on it. There are 10480142147 ways of partitioning a set of 16 objects so there is our upper bound but, in terms of our square grid, some of those ways will be geometrically impossible and we would have to discount any sets greater than 7 in size because these could not be part of a solution.

I suspect, however, that the number will still be vast and grids with unique solutions will be rare. So, to complete the construction of my puzzle, it's time for a wild card.

I shall make one or more of the numbers red. The red numbers can be used more than once. In fact, let's say that the red numbers must be used in precisely the number of sets it denotes.

Take this puzzle, for example

I need to use the red 3 in three different sets. I can't pair it with the five beneath, so it must be in sets with the numbers above, to its left, and to its right. After a bit of further deduction, we land on the unique solution:

This has potential to be far more entertaining.

Now we need to check that there are a large number of potential puzzles. Each grid square contains one of five numbers and each number is either black or red. This gives us an upper bound of ten million billion grids. A proportion of these will have solutions and a smaller quantity will have unique solutions. I won't, for the reasons stated earlier, calculate how many but I would wager there are enough to fill several newspapers daily for the rest of my life (after which I wouldn't really care).

Finally, I need a convincing sounding Japanese name. If you use Google to translate "sevens" into Japanese, it returns the answer "sebunzu". This doesn't actually mean "sevens" in Japanese. It's just phonetic. But it will do.

Here then, is the world's first Sebunzu puzzle for readers to solve. Enjoy:

from matheminutes http://ift.tt/2lr77Ey

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