A Numbered Route
2 out of 3
Can you draw a continuous line through 16 numbers on this grid so that the total of the numbers you pass through is as high as possible?
Problem
Can you draw a continuous line through $16$ numbers on this grid so that the total of the numbers you pass through is as high as possible?
You may start and finish where you like, and go horizontally, vertically or diagonally but you may only pass through a number once and the line must not cross itself at any point.
If you repeat this but only go horizontally and vertically and never go diagonally, what is the highest score then?
Getting Started
Would it help to link together groups of large numbers which are next to each other? Can you find a way to link some of these groups together?
Our highest scores are both over $100$.
Student Solutions
The solution you get for this problem depends on whether you start and end at the edge of the grid or not. Georgina from Ruyton Girls School, Melbourne, Australia found a route with a total of 124 which didn't end at the edge. She sent in this diagram to show it:
When not allowed to use diagonals, Georgina found a path with a total of 114:
Thank you Georgina! Pupils at St Thomas of Canterbury, Chester wrote to say:
When we could use diagonals we found two routes that added up to 125 and this was the highest amount that we could find. When we weren't allowed to go diagonally the highest route that we found was 115. First we highlighted the numbers 8 and 9 to show us where to concentrate our route. Some of us highlighted the numbers 0-3 so that we knew the areas to avoid. Next we began a route trying to include as many high numbers as possible, whilst trying to bypass the lower numbers. Once we had set our route we added up our numbers - we checked these a maximum of four times to ensure our answers were correct. We also found it important to check that we had only used 16 numbers (We thought we had a really high answer and then realised that we had used 18 squares).
A very clearly explained method. Unfortunately we were not able to access the picture that they sent of their route. Perhaps you can find the path they are describing?
Pupils (and staff!) at Gayhurst School in Gerrards Cross have found a highest total of 126, in two different ways. Here are their routes:
Well done to all who contributed!
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem provides a strategic and very visible context in which learners can practise addition and subtraction. They will have to work systematically using trial and improvement, and they will need to look ahead to see where their route is going. Finding a route which gives a high score is not very difficult, but finding one that gives the highest possible score with sixteen numbers is challenging.
Possible approach
Key questions
Possible extension
Learners could find other routes such as the one which gives the lowest score when they are quite certain they have found the one with the highest score. Alternatively they could make their own grid for others to try.
Possible support
Suggest using a calculator to add the numbers on the route when it has been drawn out. There are two copies of the grid on this sheet.