Sudoku is an extremely elegant puzzle, and this crucial to its appeal. The rules are simple to understand and the grid – with given numbers usually presented in a symmetric pattern – is striking. Yet perhaps Sudoku is not elegant enough. Perhaps the numbers on the starting grid are an unforgivable blemish, a needless sullying of the page.

Or so argue a group of mathematicians, who have come up with a new puzzle genre: ‘clueless Sudoku’, which are Sudoku-style puzzles with a pristine starting grid. These puzzles literally don’t have a (numerical) clue.

Below are three clueless Sudoku and an almost-clueless Killer Sudoku. Scroll down for some hints and a discussion. (Click here for a printable sheet of all the grids.)

**1) A 4x4 clueless Sudoku**

Fill the grid with the numbers from 1 to 4 such that no number appears more than once in each row or column, and that the numbers in each region add to the same sum.

(Since there are 2 regions, and the sum of all numbers in the full grid is 40, each region will sum to 20.)

**2) A 5x5 clueless Sudoku**

The rules are as above, but with digits from 1 to 5. (Thus each region must sum to 15.)

**3) A 6x6 clueless Sudoku**

The rules are as above, but with digits from 1 to 6. (Thus each region must sum to 14.)

**4) A 6x6 almost clueless Killer Sudoku**

Fill in the grid with the numbers from 1 to 6, with no number appearing more than once in each row, column or region (the six-cell blocks marked in bold). The numbers in each cage (the boxes marked by dashed lines) must sum to the total marked in that cage.

Click here for a printable sheet of all the grids.

The first three puzzles are by Gerard Butters, Frederick Henle, James Henle and Colleen McGaughey, and originally appeared in the *Mathematical Intelligencer*.

The fourth puzzle is by Philip Newman, who has recently achieved celebrity status on the cult YouTube channel *Cracking the Cryptic* for a 9x9 almost clueless Killer Sudoku, which host Simon Anthony believes “will come to be viewed as an exceptional puzzle in the history of sudoku.”

Here’s an extremely entertaining clip of Simon solving this puzzle, which explains in more detail why he believes it is such an astounding creation.

If you came here for the hint, here it is:

**Hint**

I’ll help you place the first digits in puzzle 3, in which every region sums to 14. If a region is entirely on the same row or column, then it must contain different numbers. Look at the leftmost column. It contains an entire region of three cells. The only way to sum to 14 with three different numbers is 6 + 5 + 3. Thus we know that this region comprises the numbers 3, 5 and 6. Now look at the only other three-cell region. It has one cell, marked A, in the leftmost column, which we know cannot be 3, 5 or 6. By a process of elimination the cell marked A must be 4. It can’t be 1 or 2, since otherwise there aren’t large enough numbers that will sum to 14.

I’ll be back at 5pm UK with the answers. NO SPOILERS, thanks!

UPDATE: Solutions now available here.

Click here for a printable sheet of all the grids.

*I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.*

*I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.*

*Thanks to Gerard Butters, Frederick Henle, James Henle, Colleen McGaughey and Philip Newman*