Survo puzzle

Here is a simple Survo puzzle with 3 rows and 4 columns:

Numbers 3, 6, and 8 are readily given. The task is to put remaining

numbers 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 with

fewest missing numbers. In this case columns A, B, and C

are such.

Column A is not favorable since the

sum 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 missing

numbers is 10 having only one partition 10 = 1 + 9 since the other alternatives

10 = 2 + 8 = 3 + 7 = 4 + 6 are not accepted due to numbers already present in the

table.

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

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 row

would be 30 - 7 - 6 = 17 and this allows no permitted partition. Thus we have

implying that the last number in the last row will be 30 - 7 - 9 -3 = 11:

In the first row the sum of the missing numbers is 30 - 12 - 6 = 12. Its only

possible partition is 12 = 2 + 10 and so that number 2 will be in the column C; 10 in

this position is too much for the column sum.

The solution is then easily completed as

Thus basic arithmetics and simple reasoning is enough for solving

easy 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 much

smaller than in Sudoku and Kakuro.

Mustonen, Seppo (2006): [http://solmu.math.helsinki.fi/2006/3/ "Survo-ristikot"] ("Survo puzzles"). Solmu 3/2006: 22–23.

The solving strategies are varying depending on the difficulty

of the puzzle.

In their simplest form, as in the following 2 × 3 case (degree of difficulty 0)

Survo puzzles are suitable exercises in addition and subtraction.

The open 3 × 4 Survo puzzle (degree of difficulty 150)

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 of

the numbers readily, for example, as

which makes the task almost trivial (degree of difficulty 0).

Measuring the degree of difficulty is based on

the number of 'mutations' needed by the first solver program

made by Mustonen in April 2006. This program works by using

a partially randomized algorithm.

Mustonen, Seppo (2006-06-02): [http://www.survo.fi/papers/puzzles.pdf ”On certain cross sum puzzles”]. Retrieved on 2009-08-30.

The program starts by inserting the missing numbers randomly in the

table and tries then to get the computed sums of rows and columns as close to the

true 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 end

where the discrepancy between computed and true sums cannot be diminished

systematically. In the latter case a 'mutation' is made by exchanging two or more

numbers randomly. Thereafter the systematic procedure plus mutation is repeated

until a true solution is found.

In most cases, the mean number of mutations works as a crude measure for the

level of difficulty of solving a Survo puzzle. This measure (MD) is computed as the

mean number of mutations when the puzzle is solved 1000 times by starting from

a 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:

Mustonen, Seppo (2006-09-26): [http://www.survo.fi/ristikot/vaikeusarvio.html ”Survo-ristikon vaikeuden arviointi”] (”Evaluating the degree of difficulty of a Survo Puzzle”). Retrieved on 2009-08-30.

MD

The degree of difficulty given as an MD value is rather inaccurate

and it may be even misleading when the solution is found

by clever deductions or by creative guesswork.

This measure works better when it is required that the solver

also 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 one

of them cannot converted to another by interchanging rows and

columns or by transposing when m = n.

In these puzzles the row and column sums are distinct.

The number of essentially different and uniquely solvable

m × n Survo puzzles is denoted by S(m,n).

Reijo Sund was the first to pay attention

to enumeration of open Survo puzzles. He calculated S(3,3)=38 by

studying all

9! = 362880 possible 3 × 3 tables by the standard combinatorial and data handling

program modules of Survo. Thereafter Mustonen found S(3,4)=583

by starting from all possible partitions of marginal sums and

by using the first solver program. Petteri Kaski computed

S(4,4)=5327 by converting the task into an exact cover problem.

Mustonen made in Summer 2007 a new solver program which confirms the previous

results. The following S(m,n) values have been determined by this new program:

Mustonen, Seppo (2007-10-30): [http://www.survo.fi/papers/enum_survo_puzzles.pdf ”Enumeration of uniquely solvable open Survo puzzles”]. Retrieved on 2009-08-30.

Already computation of S(5,5) seems to be a very hard task on the basis

of present knowledge.

The swapping method for solution of Survo puzzles has been created

by combining the idea of the original solver program to the observation

that the products of the marginal sums crudely indicate the positions of

the correct numbers in the final solution.

Mustonen, Seppo (2007-07-09): [http://www.survo.fi/puzzles/swapm.html ”On the swapping method”]. Retrieved on 2009-08-30.

The procedure is started by

filling the original table by numbers 1,2,...,m·n according to

sizes of these products and by computing row and column sums

according to this initial setup. Depending on how these sums deviate from

the true sums, it is tried to improve the solution

by swapping two numbers at a time. When using the swapping

method the nature of solving Survo puzzles becomes somewhat similar

to that of Chess problems. By this method it is hardly possible

to verify the uniqueness of the solution.

For example, a rather demanding 4 × 4 puzzle (MD=2050)

is solved by 5 swaps. The initial setup is

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 bookkeeping

needed 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.

The most demanding form of a quick game is available in the net

as a Java applet.

[http://www.survo.fi/java/quick5x5.html ”Survo Puzzle (5x5 quick game) as a Java applet”]. Retrieved on 2009-08-30.

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 and

by the time used for finding the final solution.

A 4x4 version of is available for iOS devices as "Hot Box".

[https://web.archive.org/web/20160325163257/https://itunes.apple.com/us/app/hot-box/id292624476?mt=8 "Hot Box, an iOS 4x4 implementation"]. Published in October 2008.

• Sudoku
• Kakuro
• Magic square

• [http://www.survo.fi/puzzles/ Survo Puzzles]: Problems and solutions

Category:Logic puzzles