Sliding TilesTask 167 ... Years 2  10SummaryThis is an annoyingly engaging task for many students which doesn't require prerequisite mathematical skills. Given any starting arrangement of tiles, the student is asked to slide the tiles to create a given result. The initial task is actually two puzzles because each tile has a letter on one side and a number on the other. The dice is provided to assist in randomising the process of placing the tiles to begin either puzzle. This task is a simpler version of a famous puzzle which has tiles numbered 1 to 15 arranged in 4 rows of 4 with one blank space. 
Materials
Content

IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
This task is totally consistent with learning to work like a mathematician because (a) it is a problem and (b) students find it interesting. Mathematicians tell us their work begins with an interesting problem so that is what makes this task the work of a mathematician, not whether it contains skill based content. The content of 'reasoning' is content enough and the focus in this case is on:
The nice thing about the task for younger children is that there is a 50:50 chance of the dice providing a starting point that will work, so the problem can provide early success. The more common form of sliding tile puzzle is 15 tiles sliding within a 4x4 frame. The user is challenged to slide the tiles into certain configurations and is also given 'impossible' challenges. Our task has been created by applying the strategy of 'try a simpler case' and it can be simplified still further by covering up the two right hand positions and only using the tiles numbered 1 to 3. Suppose now that the challenge is to finish with the sequence 1, 2, 3, blank, that is 3 __
Attempting to solve each one of those above shows that the first, second and fourth can be changed to the required solution (the first because it already is what is asked for). Trying all 24 possibilities in this way yields exactly 12 solutions. Further, if the solutions are organised into those that solve and those that don't solve students might see that each member of the 'solve set' has a partner in the 'don't solve set' which is its reflection. For example, this one does solve:
Equally, if we begin with an arrangement that doesn't solve:
Further, looking for a way to organise the 12 solutions can lead to seeing that each is obtained from the previous one by rotation. This analysis gives clues to using the 3 x 2 frame with 5 tiles. There are 6! = 6x5x4x3x2x1 = 720 ways of placing the tiles with the blank. We can expect half of these (360) to yield solutions and half not to. We can also expect that each arrangement will have a reflected partner (and that each arrangement within the 360 is like a 'rotation' of others). So, if the tiles are placed at random then there is a 50:50 chance that they can be moved into any given arrangement. Students may or may not be capable of this level of reasoning, but may still be able to solve the puzzles on the card. As they become more confident in doing so, encourage them to first orally explain, and then try to write or draw, how to do it so that someone else can learn. 
Whole Class InvestigationTasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 
With very little effort using a word processor or drawing package, you can prepare a printable version of this puzzle which the students cut to begin the investigation. The objective of doing so is to model how to work like a mathematician, in this case emphasising:
When the class is feeling confident with the problem, a web search on 'sliding tile puzzles' will reveal several software versions which provide additional challenge and tutorials for solving particular forms of them. Perhaps these will encourage your students to publish solutions and 'how to do it' articles (slideshows, videos?) for our task. If so, we would be happy to make them available through our site. (If you choose to publish elsewhere on the web, then please acknowledge our task as the source.) At this stage Sliding Tiles does not have a matching Maths300 lesson. 
Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
The Sliding Tiles task is an integral part of:
