A Survo puzzle is a kind of logic puzzle presented (in April 2006) and studied by Seppo Mustonen.[1] The name of the puzzle is associated with Mustonen's Survo system, which is a general environment for statistical computing and related areas.[2]
In a Survo puzzle, the task is to fill an m × n table with integers 1, 2, ..., m·n so that each of these numbers appears only once and their row and column sums are equal to integers given on the bottom and the right side of the table. Often some of the integers are given readily in the table to guarantee uniqueness of the solution and/or formaking the task easier.[2]
To some extent, Survo puzzles resemble Sudoku and Kakuro puzzles.However, numbers used in the solution are not restricted to 1, 2, ..., 9 and the size of puzzle grid is typically very small.Solving Survo puzzles is also related to making of magic squares.[3]
The degree of difficulty in solving Survo puzzles is strongly varying.Easy puzzles, meant for school children, are pure exercises in addition and subtraction, while more demanding ones require also good logic reasoning.The hardest Survo puzzles cannot be solved without computers.[4]
Certain properties of the Survo system like editorial computing and the COMB operation, making e.g. restricted integer partitions, support solving of Survo puzzles.
Survo puzzles have been published regularly in Finland by Ilta-Sanomat and the scientific magazine of the University of Helsinki from September 2006.Solving of Survo puzzles was one of the three main topics in the national entrance examinationof the Finnish universities in computer science (2009).[5]
Here is a simple Survo puzzle with 3 rows and 4 columns:
| A | B | C | D | |||
| 1 | 6 | 30 | ||||
| 2 | 8 | 18 | ||||
| 3 | 3 | 30 | ||||
| 27 | 16 | 10 | 25 |
Numbers 3, 6, and 8 are readily given. The task is to put remainingnumbers of 1-12 (3×4=12) to their places so that the sums are correct.
The puzzle has a unique solution found stepwise as follows:The missing numbers are 1,2,4,5,7,9,10,11,12.Usually it is best to start from a row or a column withfewest missing numbers. In this case columns A, B, and Care such.
Column A is not favorable since thesum 19 of missing numbers can be presented according to the rules in several ways(e.g. 19 = 7 + 12 = 12 + 7 = 9 + 10 = 10 + 9). In the column B the sum of missingnumbers is 10 having only one partition 10 = 1 + 9 since the other alternatives10 = 2 + 8 = 3 + 7 = 4 + 6 are not accepted due to numbers already present in thetable.Number 9 cannot be put in the row 2 since then the sum of this row would exceed the value 18. Therefore, the only choice is to start the solution by
| A | B | C | D | |||
| 1 | 6 | 30 | ||||
| 2 | 8 | 1 | 18 | |||
| 3 | 9 | 3 | 30 | |||
| 27 | 16 | 10 | 25 |
Now the column A has only one alternative 27 - 8 = 19 = 7 + 12 = 12 + 7.Number 7 cannot be in the row 1 because the sum of missing numbers in that rowwould be 30 - 7 - 6 = 17 and this allows no permitted partition. Thus we have
| A | B | C | D | |||
| 1 | 12 | 6 | 30 | |||
| 2 | 8 | 1 | 18 | |||
| 3 | 7 | 9 | 3 | 30 | ||
| 27 | 16 | 10 | 25 |
implying that the last number in the last row will be 30 - 7 - 9 -3 = 11:
| A | B | C | D | |||
| 1 | 12 | 6 | 30 | |||
| 2 | 8 | 1 | 18 | |||
| 3 | 7 | 9 | 3 | 11 | 30 | |
| 27 | 16 | 10 | 25 |
In the first row the sum of the missing numbers is 30 - 12 - 6 = 12. Its onlypossible partition is 12 = 2 + 10 and so that number 2 will be in the column C; 10 inthis position is too much for the column sum.
| A | B | C | D | |||
| 1 | 12 | 6 | 2 | 10 | 30 | |
| 2 | 8 | 1 | 18 | |||
| 3 | 7 | 9 | 3 | 11 | 30 | |
| 27 | 16 | 10 | 25 |
The solution is then easily completed as
| A | B | C | D | |||
| 1 | 12 | 6 | 2 | 10 | 30 | |
| 2 | 8 | 1 | 5 | 4 | 18 | |
| 3 | 7 | 9 | 3 | 11 | 30 | |
| 27 | 16 | 10 | 25 |
Thus basic arithmetics and simple reasoning is enough for solvingeasy Survo puzzles like this one.
The rules of Survo puzzles are simpler than those of Sudoku.The grid is always rectangular or square and typically muchsmaller than in Sudoku and Kakuro.[6]
The solving strategies are varying depending on the difficultyof the puzzle.[6] In their simplest form, as in the following 2 × 3 case (degree of difficulty 0)
| 3 | 9 | ||
| 6 | 12 | ||
| 9 | 7 | 5 |
Survo puzzles are suitable exercises in addition and subtraction.[6]
The open 3 × 4 Survo puzzle (degree of difficulty 150)
| 24 | |||||
| 15 | |||||
| 39 | |||||
| 21 | 10 | 18 | 29 |
where none of the numbers are readily given, is much harder.Also it has only one solution.
The problem can be simplified by giving some ofthe numbers readily, for example, as
| 7 | 5 | 24 | |||
| 1 | 8 | 15 | |||
| 11 | 39 | ||||
| 21 | 10 | 18 | 29 |
which makes the task almost trivial (degree of difficulty 0).[6]
Measuring the degree of difficulty is based onthe number of 'mutations' needed by the first solver programmade by Mustonen in April 2006. This program works by usinga partially randomized algorithm.[7]
The program starts by inserting the missing numbers randomly in thetable and tries then to get the computed sums of rows and columns as close to thetrue ones as possible by exchanging elements in the table systematically.This trial leads either to a correct solution or (as in most cases) to dead endwhere the discrepancy between computed and true sums cannot be diminishedsystematically. In the latter case a 'mutation' is made by exchanging two or morenumbers randomly. Thereafter the systematic procedure plus mutation is repeateduntil a true solution is found.In most cases, the mean number of mutations works as a crude measure for thelevel of difficulty of solving a Survo puzzle. This measure (MD) is computed as themean number of mutations when the puzzle is solved 1000 times by starting froma randomized table.The distribution of the number of mutations comes close to a geometric distribution.
These numeric values are often converted to a 5-star scale as follows:[8]
MD
| 0 - 30 | ||
| 31 - 150 | ||
| 151 - 600 | ||
| 601 - 1500 | ||
| 1500 - |
The degree of difficulty given as an MD value is rather inaccurateand it may be even misleading when the solution is foundby clever deductions or by creative guesswork.This measure works better when it is required that the solveralso proves that the solution is unique.
A Survo puzzle is called open, if merely marginal sums are given.Two open m × n puzzles are considered essentially different if oneof them cannot converted to another by interchanging rows andcolumns or by transposing when m = n.In these puzzles the row and column sums are distinct.The number of essentially different and uniquely solvablem × n Survo puzzles is denoted by S(m,n).[7]
Reijo Sund was the first to pay attentionto enumeration of open Survo puzzles. He calculated S(3,3)=38 bystudying all9! = 362880 possible 3 × 3 tables by the standard combinatorial and data handlingprogram modules of Survo. Thereafter Mustonen found S(3,4)=583by starting from all possible partitions of marginal sums andby using the first solver program. Petteri Kaski computedS(4,4)=5327 by converting the task into an exact cover problem.
Mustonen made in Summer 2007 a new solver program which confirms the previousresults. The following S(m,n) values have been determined by this new program:[9]
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
| 2 | 1 | 18 | 62 | 278 | 1146 | 5706 | 28707 | 154587 | 843476 | |
| 3 | 18 | 38 | 583 | 5337 | 55815 | 617658 | ||||
| 4 | 62 | 583 | 5327 | 257773 | ||||||
| 5 | 278 | 5337 | 257773 | |||||||
| 6 | 1146 | 55815 | ||||||||
| 7 | 5706 | 617658 | ||||||||
| 8 | 28707 | |||||||||
| 9 | 154587 | |||||||||
| 10 | 843476 |
Already computation of S(5,5) seems to be a very hard task on the basisof present knowledge.
The swapping method for solution of Survo puzzles has been createdby combining the idea of the original solver program to the observationthat the products of the marginal sums crudely indicate the positions ofthe correct numbers in the final solution.[10] The procedure is started byfilling the original table by numbers 1,2,...,m·n according tosizes of these products and by computing row and column sumsaccording to this initial setup. Depending on how these sums deviate fromthe true sums, it is tried to improve the solutionby swapping two numbers at a time. When using the swappingmethod the nature of solving Survo puzzles becomes somewhat similarto that of Chess problems. By this method it is hardly possibleto verify the uniqueness of the solution.
For example, a rather demanding 4 × 4 puzzle (MD=2050)
| 51 | |||||
| 36 | |||||
| 32 | |||||
| 17 | |||||
| 51 | 42 | 26 | 17 |
is solved by 5 swaps. The initial setup is
| Sum | OK | error | |||||
| 16 | 15 | 10 | 8 | 49 | 51 | -2 | |
| 14 | 12 | 9 | 4 | 39 | 36 | 3 | |
| 13 | 11 | 6 | 3 | 33 | 32 | 1 | |
| 7 | 5 | 2 | 1 | 15 | 17 | -2 | |
| Sum | 50 | 43 | 27 | 16 | |||
| OK | 51 | 42 | 26 | 17 | |||
| error | -1 | 1 | 1 | -1 |
and the solution is found by swaps (7,9) (10,12) (10,11) (15,16) (1,2).In the Survo system, a sucro /SP_SWAP takes care of bookkeepingneeded in the swapping method.
Solving of a hard Survo puzzle can take several hours.Solving Survo puzzles as quick games offers another kind of challenges.[4] The most demanding form of a quick game is available in the netas a Java applet.[11] In this quick game, open 5 × 5 puzzles are solved by selecting (or guessing)the numbers by mouse clicks. A wrong choice evokes a melodic musical interval.Its range and direction indicate the quality and the amount of the error.The target is to attain as high score as possible.The score grows by correct choices and it is decreased by wrong ones andby the time used for finding the final solution.
A 4x4 version of is available for iOS devices as "Hot Box".[12]