So you think you are clever, right? Then here is your chance to pit your brain against some of the world's hardest logic puzzles ever created. After having created number puzzles like Calcudoku and Killer Sudoku for many years, I decided to try and find the most challenging ones out there. Every once in a while I added a new type of puzzle, until I ended up with a list of 10.

In the following list you will find both familiar puzzles and games such as Sudoku and Calcudoku as well as lesser known ones such as the Bongard Problem and Fill-a-Pix. Some of these puzzles can be solved right on this page while others can be downloaded or reached elsewhere. All of them, however, are promised to test your solving skills to the absolute limit and keep you busy for hours, if not days.

Find an even harder puzzle? Be sure to let me know! For more information about this project and other logic puzzles visit my website Calcudoku.org

## 1. The World's Hardest Sudoku

Sudoku is easily the most played and most analysed puzzle in the world, so coming up with the hardest one is no mean feat. In 2012, Finnish mathematician Arto Inkala claimed to have created the "World's Hardest Sudoku".

According to the British newspaper The Telegraph, on the difficulty scale by which most Sudoku grids are graded, with one star signifying the simplest and five stars the hardest, the above puzzle would "score an eleven". More information on how Inkala's puzzles are rated is on his website.

## 2. The Hardest Logic Puzzle Ever

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.

American philosopher and logician George Boolos invented the above riddle, published in the Harvard Review of Philosophy in 1996, and called it "The Hardest Logic Puzzle Ever". The original article can be downloaded here. You can read about making this puzzle even harder on the Physics arXiv Blog.

## 3. The World's Hardest Killer Sudoku

A Killer Sudoku is very similar to a Sudoku, except that the clues are given as groups of cells + the sum of the numbers in those cells. From a large number of highest rated puzzles at Calcudoku.org, I measured what percentage of puzzlers solved them on the day they were published. Easily the hardest was the Killer Sudoku shown above, published on the 9th of November 2012. You can solve this puzzle right here.

## 4. The Hardest Bongard Problem

This type of puzzle first appeared in a book by Russian computer scientist Mikhail Moiseevich Bongard in 1967. They became more widely known after Douglas Hofstadter, an American professor of cognitive science, mentioned them in his book "Gödel, Escher, Bach". To solve the above puzzle, published on Harry Foundalis' website, you have to find a rule that the 6 patterns on the left hand side conform to. The 6 patterns on the right do not conform to this rule. For example, the first problem on this page has as a solution: all patterns on the left are triangles.

## 5. The Hardest Calcudoku Puzzle

A Calcudoku is similar to a Killer Sudoku, except that (1) any operation can be used to compute the result of a "cage" (not only addition), (2) the puzzle can be any square size, and (3) the Sudoku rule of requiring the numbers 1..9 in each 3×3 set of cells does not apply. Calcudoku was invented by Japanese maths teacher Tetsuya Miyamoto, who called it "Kashikoku naru" ("smartness").

Identified in the same way as the Killer Sudoku presented in this article, the hardest Calcudoku was a 9×9 puzzle published on April 2, 2013, which only 9.6% of the regular puzzlers at Calcudoku.org managed to solve. You can give it a try right here. If you're not up for solving it yourself, check out this step-by-step solving analysis by "clm".

## 6. The Hardest "Ponder this" Puzzle

Design a storage system that encodes 24 information bits on 8 disks of 4 bits each, such that:

1. Combining the 8*4 bits into a 32 bits number (taking a nibble from each disk), a function f from 24 bits to 32 can be computed using only 5 operations, each of which is out of the set {+, -, *, /, %, &, |, ~} (addition; subtraction, multiplication; integer division, modulo; bitwise-and; bitwise-or; and bitwise-not) on variable length integers. In other words, if every operation takes a nanosecond, the function can be computed in 5 nanoseconds.

2. One can recover the original 24 bits even after any 2 of the 8 disks crash (making them unreadable and hence loosing 2 nibbles)

IBM Research has been publishing very challenging monthly puzzles since May 1998 on their Ponder this page. Judging from the number of solvers for each, the hardest number puzzle is the one shown above, published in April 2009. If you need some clues visit this page.

## 7. The Hardest Kakuro Puzzle

Kakuro puzzles combine elements of Sudoku, logic, crosswords and basic maths into one. The object is to fill all empty squares using numbers 1 to 9 so the sum of each horizontal block equals the clue on its left, and the sum of each vertical block equals the clue on its top. In addition, no number may be used in the same block more than once.

Those in the know tell me that the Absolutely Nasty Kakuro Series by Conceptis Puzzles has the world's hardest Kakuro puzzles. Gladly, the guys at Conceptis have produced the above even nastier Kakuro specimen, especially for this article. Play this puzzle online here.

## 8. Martin Gardner's Hardest Puzzle

A number's persistence is the number of steps required to reduce it to a single digit by multiplying all its digits to obtain a second number, then multiplying all the digits of that number to obtain a third number, and so on until a one-digit number is obtained. For example, 77 has a persistence of four because it requires four steps to reduce it to one digit: 77-49-36-18-8. The smallest number of persistence one is 10, the smallest of persistence two is 25, the smallest of persistence three is 39, and the smaller of persistence four is 77. What is the smallest number of persistence five?

Martin Gardner (1914-2010) was a popular American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature, philosophy, scientific scepticism and religion (Wikipedia). In his book The Colossal Book of Short Puzzles and Problems puzzles in many categories are listed in order of difficulty. The above is the hardest puzzle from the "Numbers" chapter.

## 9. The Most Difficult Go Problem Ever

Go is a board game for two players that originated in China more than 2,500 years ago. The game is noted for being rich in strategy despite its relatively simple rules (Wikipedia). The above problem is considered to be the hardest ever and is said to have taken 1000 hours to solve by a group of high level students. Solutions and many references can be found on this page.

## 10. The Hardest Fill-a-Pix Puzzle

Fill-a-Pix is a Minesweeper-like puzzle based on a grid with a pixilated picture hidden inside. Using logic alone, the solver determines which squares are painted and which should remain empty until the hidden picture is completely exposed. Advanced logic Fill-a-Pix such as the one above contain situations where two clues simultaneously affect each other as well as the squares around them making these puzzles extremely hard to solve.

Fill-a-Pix was invented by Trevor Truran, a former high-school maths teacher and the editor of Hanjie and several other famed British magazines published by Puzzler Media. For Fill-a-Pix solving rules, advanced solving techniques and more about the history of this puzzle check the Get started section on conceptispuzzles.com. This ultra-hard puzzle was generated by Conceptis especially for this article and can be played online here.

*This article originally appeared on Conceptis Puzzles and is reproduced here with kind permission. Conceptis is the leading supplier of logic puzzles to printed and electronic gaming media all over the world. On average, more than 20 million Conceptis puzzles are solved each day in newspapers and magazines,online and on mobile platforms across the world.*

*Patrick Min is a freelance scientific programmer. He specialises in geometry software, but has worked in many other areas, such as search engine technology, acoustic modelling, and information security. He has published several papers and open/closed-source software across these subjects. Patrick holds a Master's degree in Computer Science from Leiden University, the Netherlands, and a Ph.D. in Computer Science from Princeton University. He is also a puzzle enthusiast, devising maths puzzles for his father since the age of 7. This continues to this date, with dad solving his son's Calcudoku puzzles. Patrick lives in London.*

*Top art by David Masters under Creative Commons licence.*