This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
This activity investigates how you might make squares and pentominoes from Polydron.
If you had 36 cubes, what different cuboids could you make?
Published September 2011,July 2011.
In this article, members of the NRICH team explain the process they go through when they choose solutions to publish on the site.
Each time you visit the site you will see that there are some "live problems" which are open for solutions. This means we are inviting students to send in thoughts, ideas, workings out and solutions, and then we'll publish them after a suitable time so that our archives are complete with both problems and solutions. On average problems remain live for about six weeks. They have a "Submit a
solution" link to the left of the Problem title.
To check which problems have been recently solved you can visit the "Recently solved" pages.
As well as wanting to showcase the hard work and effort that has been put into solutions, we know that lots of students, teachers and parents find it useful to look at the solutions to the problems in our archive.
Students might want to check that their answers and methods agree with what someone else has done.
Teachers might want to show their classes different approaches to solving a problem. Parents might want to see how problems can be solved in order to support their children.
We publish solutions that are clearly explained so that students, teachers and parents can understand them.
We receive solutions in a variety of formats. Here are a few examples:
Emily's and Niharika's handwritten and scanned solutions to Opposite Vertices
The spreadsheet that Susan sent for her solution to Constructing Triangles
George and Sophie's photo of their completed Fraction Jigsaw
The video that Martha made in response to Perception versus Reality
The first decision to make is whether a solver has sent us an "answer" or a "solution". An answer is brief and just contains answers to the questions we've posed. A solution may not be complete, but will include the solver's ideas, why they chose a certain method, and explanations of what they noticed. We want someone looking at the published solution to a problem to get some understanding about
the sort of thought processes that lead to the answer, so they can learn from someone else's work and become a better problem-solver.
Secondly, we need to make sure that the solution is mathematically correct. If there are one or two minor errors, we will correct them and publish the solution, but if a solution showed a major misconception or the solver didn't really explain their method, we wouldn't publish their solution. Of course if a solution began very clearly but had lots of errors towards the end, we might publish
just the first part of that solver's work.
Once we've eliminated all the submitted solutions that are just answers without explanation, we then think about how to organise the solutions.
At Stages 1 and 2, to begin with we look out for solutions that perhaps tackle just the first part of the problem and we often publish these at the top of the page. We would then choose solutions that address the question more fully and we often order them to reflect increasingly sophisticated methods. We don't want to put anyone off having a go at the problem, which is why
solutions using higher-level mathematics come nearer the bottom of the page.
For example, in the published solutions to the Stage 1 task, The Add and Take-away Path, Jack and Ellie answer the first and second parts of the problem, giving us the scores for each grid. Then, the other solutions go on to explain why this always happens.
In a similar way, the solutions for the Stage 2 problem Three Dice begin with some observations and the suggestion of a "rule". Several learners then offer very clear explanations for why this always happens. Finally, we also include solutions which look at more than three dice and go on to generalise the problem.
For the Stage 3 and 4 solutions, we start by looking for solutions which make a good start on the problem but perhaps don't answer the whole question. For example, quite a few problems involve numerical relationships which can be analysed using algebra, so we regularly receive solutions from younger students who made lots of progress with the numerical part of the problem but don't always
proceed to the algebraic parts. We then pick out the solutions which develop the next part of the problem, and feature those next. If someone has used some particularly advanced mathematics (perhaps using ideas not normally met until Stage 5), we usually publish their solution nearer the bottom of the page. This means that anyone reading the solution can read as far as they understand and then
take a look at what might come next as they learn new mathematical techniques.
Take a look at the solutions to the March 2011 problem Multiplication Arithmagons. Lots of students came up with the method described by Alexander, so we chose to feature his very clear description and mention everyone else who had worked in the same way. Fionn's algebraic approach came next, and Francesco's solution added some extra
detail (each solution could be multiplied by -1). At the bottom of the page, we provided links to those solutions we had received as uploaded attachments.
At stage 5 many of the problems are reasonably easy to start but somewhat more difficult to complete. We very often receive partial solutions to the problems and use these to piece together a full solution. We will correct minor mistakes and very often paraphrase solutions. As with other stages, we want to know what was going through your mind as you constructed your solution, as this is
useful to share with other solvers, as well as making the solutions more entertaining! One particular feature of many stage 5 problems is that they involve a great deal of mathematical symbolism. Taking care to format this appropriately will increase your chances of the solution appearing on the site. You can see how the full solution is constructed collaboratively in the solution to the problem
Black Box. Sometimes, if a solution is good enough, we simply publish the entire contribution as an additional download from the solutions page: see Escape from Planet Earth as an example of this.
If after a suitable time we haven't received a solution to a problem, we mark it as a "Toughnut". You can submit solutions to toughnuts at any time, and then after a few months the team will revisit the problem to see if any solutions have been received. Once a solution is received, we publish it and remove the problem from the toughnuts list. If you submit a solution to the toughnut, be
patient - it may not appear on the site for quite a while, as we usually revisit the toughnuts during the summer months.
Finally, here are five tips for students on how to get your solutions published:
1) Check your solution carefully, and perhaps ask someone else to look over it to spot any mistakes you might have missed.
2) Try to explain simply and clearly what you did to solve the problem. It doesn't have to be a long explanation, but we like solutions that share the mathematical journey of the solver.
3) If you made any diagrams or used a spreadsheet to help you, upload it with your solution. Make sure you explain clearly what you are referring to if you talk about your diagram or spreadsheet in your solution.
4) If you upload your solution as a document, keep it as simple as possible. If you send a solution with lots of formatting, your solution might not appear on our screens as you intended. If you use a less common file type, we might not be able to read it easily, but .doc and .pdf files are fine.
5) Read through other people's solutions to previous problems on the site. This will give you a flavour for the sorts of solutions we publish, and help you to develop your own mathematical style for writing up your work. Learning how to communicate mathematically is an important part of learning to be a mathematician!
When we publish solutions, we don't publish full names or ages, so if you submit your solution as a pdf, just write your first name.