A Numberskull puzzle consists of a grid with 6 = signs and 6 whole number results already on it, together with 9 given whole numbers, all different. You simply have to fill in the grid by placing each of the nine numbers into the nine white squares, one in each, and placing any one of the four basic arithmetic operators (+, -, x, ÷) into the grey squares, so that 6 correct equations are formed (3 in left-to-right rows, 3 in top-to-bottom columns).
Numberskull Example
Nine numbers: 2,3,6,7,11,31,40,44,49
Numberskull Solution
Note that the first operation is always evaluated first and its result is used in the second operation to produce the overall result. So, 3 + 4 x 2 means (3 + 4) x 2, which simplifies to 7 x 2 and equals 14. It does not mean 3 + (4 x 2) which simplifies to 3 + 8 and equals 11.
There is a unique solution. This can be found a step at a time, with no guessing required, by using arithmetic to determine some of the possible ways of achieving the results with the given numbers and, where necessary, by then applying logic to determine more about the positions of the numbers.
Using the system
When Numberskull first opens, one of the built-in Numberskull puzzles is loaded ready for you to solve.
The nine numbers are displayed in the bottom-right corner of the puzzle, with a light-blue background. These also act as buttons. There are also four light-blue buttons below the grid, displaying each of the four operators.
To place a number or an operator on the grid, first press the relevant light-blue button. The background of the button turns green to indicate that it is selected. Once any number or operator is selected, pressing in the desired cell of the grid places the number or operator in that cell (but remember that numbers can only be placed in white cells and operators in grey cells). Once a cell has been pressed, the number or operator is no longer selected and the background reverts to light blue.
If no number or operator is selected (green background), pressing on a cell containing one of the nine numbers or operators removes it from the grid.
Note: if you place the same number in two cells, the system warns you of this in the text field below the grid.
At any time, you can check your progress by pressing on the Check button. This looks at each of the six lines/equations and checks the correctness (in terms of the solution) of each number and operator in the line. If all numbers or operators have been placed and are correct, the background of the line's result turns green. If one or more of the numbers or operators is incorrect, the result's background turns red. If the line is incomplete, but all the numbers or operators already placed are correct, then the background turns yellow. So, six green result backgrounds confirms that you have completed the Numberskull puzzle.
If you find that you can't solve the puzzle and wish to see the solution, press the Solve button. The solution is presented, together with the approach taken to solve it (see later), which is displayed in the text field.
To load a new Numberskull, press the Load button. This loads a Numberskull of the selected level (1 or 2, or either - see later for definitions of these). To change the level, press on the More button, then select the level as 1 or 2, or, if you wish to have either loaded, select the ? option (which is the default option when Numberskull is first loaded). Once you have selected one of these, press the Back button to return to the main screen.
Solution Techniques
The idea seems very simple, and the unwary puzzler might think that it is easy to solve. However, it soon becomes apparent that many of the target numbers can be achieved in very many ways. For example, the target 52 in the example puzzle can be achieved in 19 ways with the given numbers:
The target 2 can be achieved in 55 ways, while the target 40 can be achieved in 49 ways. We find that, on average, each target number can be achieved in about 20 ways at the beginning of a puzzle in which no numbers or operators are already fixed on the grid. What might have seemed to be a simple puzzle now starts to look extremely difficult.
However, Numberskull puzzles are designed to ensure that they have unique solutions, and that they can be solved a step at a time by following certain techniques.
Although you might use various techniques to solve them, Numberskull puzzles are designed to be solved using a set of eight techniques. At each step, you should try to apply these in order, and find one that takes you a step further towards the unique solution. The first four techniques involve looking only at individual lines (equations); the remainder involve looking at number squares across the grid and can include considering the implications of the intersection of row and column equations. For example, if a square can take 4, 7 or 36 according to its row equation, and 4, 36, 39 or 42 according to its column equation, the square can therefore take only those numbers that are common to both, namely 4 or 36.
Technique 1:
Look for an incomplete equation for which there is only one way of achieving the result. The complete line can be fixed to the appropriate values.
Example: Given the nine numbers 2,3,6,7,11,31,40,44,49, suppose the result to be achieved is 38. This can be achieved in only one way:
Technique 2:
Look for an incomplete equation where there are two or more ways of achieving the result, but in all of these cases one square takes the same number. That square MUST therefore contain that number, so that number can be placed in that square on the grid.
Example: Given the nine numbers 1,5,6,7,15,17,29,40,45, suppose the result to be achieved is 96. This can be achieved in several ways:
However, the 3rd number is always a 6, so the square for the 3rd number must contain the 6.
Technique 3:
Look for an incomplete equation where there are two or more ways of achieving the result, but in all of these cases there are two squares that can each take only the same two numbers. Between them, these two squares MUST take these two numbers. Even though we cannot, at this stage, say which of these two eventually takes which number, we now know:
- These two squares can take only these two numbers
- All other squares cannot take these two numbers
This information can help us towards our overall solution.
Example: Given the nine numbers 1,2,3,4,5,6,8,13,33, suppose the result to be achieved is 85. This can be achieved in four ways only:
However, the 1st and 2nd numbers are always either a 4 or 13, so between them they must contain the 4 and 13. 4 and 13 can now be eliminated as possible numbers for all other squares.
Technique 4:
Look for an incomplete equation where there are three or more ways of achieving the result, but the same three numbers are used in all the possible ways. Between them, these three squares MUST take these three numbers. Even though we cannot, at this stage, say which of these three eventually takes which number, we now know:
- These three squares can take only these three numbers
- All other squares cannot take these three numbers
This information can help us towards our overall solution.
Example: Given the nine numbers 1,4,9,10,15,27,31,39,45, suppose the result to be achieved is 83. This can be achieved in four ways only:
Between them, the three number squares can take only the values 1, 39 and 45, so these numbers can be eliminated as potential values from all other squares. The potential values for the 1st number square of this line can be constrained to 39 or 45, while the 2nd and 3rd can both be constrained to 1, 39 or 45.
Technique 5:
Look for an empty number square where the possible values for its row equation have only one number in common with the possible values for its column equation and where the row or column (or maybe both) has only one way of achieving its result with this common value in this number square. The number square can be fixed to this common value, and the row or column with the resulting unique equation (or both) can be completed.
Example: Given the nine numbers 2,3,5,6,16,20,33,47,49, suppose the result to be achieved in the bottom row is 89 and the result for the middle column is 92. The 89 can be achieved in several ways:
as can the 92:
The potential values for the shared square are 2,20 or 47 for the row (its 2nd number) and 2,6 or 49 for the column (its 3rd number). As these have only one number in common, 2, this square must take 2. Only one of the equations for the column has a 2 in this position
so the column can be fixed to these values. However, there are two equations with 2 in the 2nd place for the row, so this cannot be fixed.
Technique 6:
Look for an empty number square where the possible values for its row equation have only one number in common with the possible values for its column equation but where this common value does not yield a unique result in either direction. The number square can be fixed to this common value.
Example: Given the nine numbers 2,3,5,10,12,20,26,31,45, suppose the result to be achieved in the top row is 71 and the result for the right column is 82. The 71 can be achieved in several ways:
as can the 82:
The potential values for the shared square are 5, 31 or 45 for the row (its 3rd number) and 2, 10 or 31 for the column (its 1st number). As these have only one number in common, 31, this square must take 31. However, in neither the row nor the column is there a unique equation with 31 in the shared position, so no complete line can be fixed, but the square can be fixed to 31.
Technique 7:
Look for two empty number squares from anywhere on the grid that can each take only the same two numbers. Between them, these two squares MUST take these two numbers. Even though we cannot, at this stage, say which of these two eventually takes which number, we now know:
- These two squares can take only these two numbers
- All other squares cannot take these two numbers
When looking for squares that can take only two numbers, we can look either at the possible numbers for the square according to its row or column individually, or at the possible numbers that result from considering the intersection of the row and column. Overall, you should consider the possible numbers once intersections have been considered, but sometimes this is not feasible and you may have to consider just the possible values for its row or column.
Example: Given the nine numbers 2,4,6,9,15,18,22,32,44, suppose the result to be achieved in one line is 73 and the result in another line is 79. The 73 can be achieved in several ways:
as can the 79:
So, even without considering any interactions of rows and columns, we can see that the 3rd number in both lines can be only a 9 or 15. Therefore, we can eliminate 9 and 15 as potential numbers for all other squares.
Given the nine numbers 3,6,9,10,12,14,21,32,35, suppose the result to be achieved in the top row is 80 and the result to be achieved in the left column is 91. The 80 can be achieved in several ways:
as can the 91:
Looking at the 3rd number in the column (91), we see that it can take only 14 or 35, but no other squares can take only these two numbers when the lines are considered individually. However, if we look at the intersection of this particular row and column (the first number in both cases), we see that the only numbers common to both are 14 and 35, so this square can also take only 14 or 35. We have therefore identified two squares that can take only 14 or 35, so can eliminate these two numbers as possible values from all other squares.
Technique 8:
Look for three empty number squares from anywhere on the grid that, between them, can take only the same three numbers. Between them, these three squares MUST take these three numbers. Even though we cannot, at this stage, say which of these three eventually takes which number, we now know:
- These three squares can take only these three numbers
- All other squares cannot take these three numbers
When looking for squares that can that form one of the set of three, we can look either at the possible numbers for the square according to its row or column individually, or at the possible numbers that result from considering the intersection of the row and column. Overall, you should consider the possible numbers once intersections have been considered, but sometimes this is not feasible and you may have to consider just the possible values for its row or column.
Example: Given the nine numbers 4,14,19,20,23,27,35,40,42, suppose the result to be achieved in one line is 80 and the result to be achieved in another line is 99. The 80 can be achieved in several ways:
as can the 99:
The first two numbers for the 99 can take only 4, 19 or 20, while the last number for the 80 can be only 4 or 20. So, between these three squares, they must take the 4, 19 and 20, so these numbers can be eliminated as possible numbers for all other squares.
The above example did not necessitate looking at the interaction of a row and column, only at the possible numbers for individual lines. However, it may be necessary to look at the interactions in some cases when using this technique, as in the following example.
Example: Given the nine numbers 1,4,5,9,13,25,29,32,50, suppose the result to be achieved in the top row is 71 and the result for the right column is 99 (so they intersect at the 3rd number of the row and the 1st number of the column. The 71 can be achieved in several ways:
as can the 99:
Looking at the row, we can see that the first two squares can each take only 4, 25 or 50, whereas the 3rd square can take 4, 25, 50 or 29.
Looking at the column, we see that the 1st square can take only 4, 25 or 32. This means that when we consider the interaction of the row and column, the shared square can take only 4 or 25.
Altogether, then, the three squares of the top row can, between them, take only 4, 25 or 50, so these numbers can then be eliminated as possible numbers for all other squares.
Solution codes
When the Solve button is pressed, as well as completing the puzzle, the system also displays a code in the text field. This code represents the approach to solving the puzzle using the 8 techniques, attempted in order, up to and including the placement of the 6th number.
So, for the example Numberskull, we can provide the following solution code defining the way to solve this:
C3(T1),6(T2:R1),R1+C2(T5)
This states that we first complete column 3 using technique T1, then fix the 6 using T2 operating on row 1, then complete row 1 and column 2 using T5.
The technique is always shown in brackets after the lines or numbers that were fixed or constrained (see next). In the case of T2, the single line involved is shown followed by a colon. For T5, if only one line is completed, the other line involved is also shown in this way.
If the solution code contains numbers between a pair of | symbols, this means that the squares containing these numbers were constrained to these numbers at that step and the numbers eliminated from other squares. For T3 and T4 the lines involved are always obvious, while for T7 and T8 they are not considered relevant.
So, in the solution code
|4,36,49|(T4),R1(T1),C2(T1),C1(T1)
the first step uses Technique 4 to constrain the squares containing the 4, 36 and 49.
House rules
The brief outline of Numberskull given at the beginning would cover puzzles with negative numbers, results of over 1000, and so on. We employ a number of ‘house rules’ that constrain the puzzles allowed in various ways, with the intention of making them more appealing and easier, including:
- The results should all be whole numbers in the range 1-100 (numbers people are generally most familiar with). Duplicates are permissible.
- The 9 given numbers, all different, should be whole numbers in the range 1-50.
- Up till, and including, the step in which the 6th number square is fixed, no alternatives should be found when employing the 8 techniques in order as described. So, for example, if Technique 1 is used and a line is found that has a unique solution so can be fixed, there should be no other line that can be fixed using Technique 1 at that step. Or, for example, if Technique 1 fails and Technique 2 is being attempted, only one number square should exist that can be fixed by considering only its row or column. Once 6 squares have been fixed, the solution generally becomes trivial and alternatives are allowed (and become inevitable).
- Any line that has more than 6 ways of achieving the result during a step should play no part in that step, either individually or to determine the potential values at an intersection of row and column. The squares within such a line may, of course, take part as a member of another line. This means that you do not need to carry out a full analysis for all lines in order to determine the intersection, if required, but can focus on lines with 6 or fewer potential solutions.
Numberskull grading
The puzzles are graded as Level 1 or Level 2. Level 1 means that only techniques that require you to consider individual lines are required, i.e. Techniques 1-4. Level 2 puzzles will include the use of at least one higher-level technique (5-8).
Numberskull was developed by Dr. David Wolstenholme. Numberskull is a registered trademark of TopAccolades Limited. More information can be found at www.numberskull.com.
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