Challenger number game

Figure 1 provides a discrete histogram of the data in Table 1. Figure 1 Absolute frequency of the number of solutions per trial in trials. The largest absolute frequency is 21; the smallest, 1. The largest number of solutions in a trial was ; the smallest, 1. I suspect they might have something to do with the number of partitions of the row, column, and diagonal sums.

Indeed, if such a sum were, say, 25, then there would be. The first of these partitions, namely,. We need to restrict the partitions to just those containing four integers. This much shorter list is then. Note that each of these 23 partitions is sorted, from smallest to largest integer. These integers might appear permuted in the solution to a Challenger puzzle in which 25 were a row, column, or diagonal sum.

The combinatorics of this approach do not look fruitful. The code I used to generate a puzzle appears in Table 4. Table 4 Code for generating random Challenger puzzle.

The eight lists of four pairs at the top of the table represent valid configurations for the four numbers to be revealed. I noticed that the puzzles in my local newspaper had exactly one such number in each row, column, and diagonal. However, on the internet, I found puzzle generators that did not adhere to this restriction. The eight lists in Table 4 are exhaustive - they represent the only ways to satisfy the constraint stated.

One such configuration is selected at random. This matrix contains the four displayed digits, and the remaining twelve entries are the variables. From , the row, column, and diagonal sums can be computed, and from , these sums can be expressed as ten equations in the twelve unknowns. Maple's solve and isolve integer-solve commands provide the same solution, and for the cases I examined, each solution contained not just two indeterminates, but three.

A typical such solution vector appears in Table 5, where we have called the three indeterminates. Table 5 Typical solution vector whose components are.

Each component of this vector must be an integer in the interval , a condition that generates 24 linear inequalities. Maple does solve this set of inequalities, but it takes a noticeable amount of time, and the solutions themselves can contain further inequalities.

For these reasons, a "brute-force" solution was adopted for vectors like the one in Table 5. All possible integers in the range were substituted for each of , , and ; then the twelve components of the resulting vector were examined for the maximum and minimum.

If the maximum was no larger than 9, and the minimum no smaller than 1, the solution was accepted as valid.

In fact, to test the validity of such solutions, the row, column, and diagonal sums were checked and found to be correct. I imagine that a puzzle with nearly solutions would be more difficult to solve that one with just one or two solutions.

That would probably be because most empty squares would admit so many different integers. Table 6 contains a puzzle with some solutions.

Table 6 Challenger puzzle with solutions: in the center, the starting array; on the left, the "puzzle"; on the right, the possible integers in some solution of the puzzle. At first, I thought the blue cells in Table 6 were a surprise. These cells contain the displayed integers for the puzzle, and in every one of the solutions, these cells contain just the single displayed integer. However, the uncolored cells are the ones corresponding to the variables , so it is only in these uncolored cells that multiple integers can appear.

The complete code for creating, displaying, and solving these Challenger puzzles is available behind the "New Puzzle" button, and the Code Edit Region icon. Tags are words are used to describe and categorize your content. Combine multiple words with dashes - , and seperate tags with spaces. You must be logged in to your Twitter account in order to share.

Click the button below to login a new window will open. My Profile points. Log out. New Games Most Popular Games. Adjust game screen size. Game controls. Adjust or maximize. Numbers Challenge 2, play times. Added to your profile favorites. Human validation. Thank you, your vote was recorded and will be displayed soon. Tags All tags. Add this game to your web page Share on Website.

Add this game to your web page Numbers Challenge. Join other players talking about games. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent.

You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience. Necessary Necessary. Necessary cookies are absolutely essential for the website to function properly.