Thread: numbering squares

The numbers 1 to 25 are to be placed on a black and white 5x5 grid so that each number, except 1, is next to (horizontally or vertically), the number one less than itself. eg.



a) Explain why number 1 must be placed on a black square
b) Explain why the numbers 1, 3, 5, 7, 9 can not all be placed on one long diagonal
c) Show that in a completed numbering the sum of the numbers on a long diagonal must be at least 33 and that 33 can occur.

Any help is appreciated!
Thanks!

Originally Posted by shosho

The numbers 1 to 25 are to be placed on a black and white 5x5 grid so that each number, except 1, is next to (horizontally or vertically), the number one less than itself. eg.



a) Explain why number 1 must be placed on a black square

The sequience of squares that have 1, 2, 3, ... , 25 form a path of adjacent
squareswhich covers the grid such that consecutive numbers are on different
coloured squares. There are 13 black squares and 12 white squares, so the path
must start and end on black squares. Therefor 1 is on a black square.

RonL

Originally Posted by shosho

b) Explain why the numbers 1, 3, 5, 7, 9 can not all be placed on one long diagonal

Because the chain of squares labled 1,2,3,4,5,6,7,8,9 will divide the grid
into two seperate parts, so as the path must continue initialy in one of these
halves, it can never cross again into the other half so all the squares cannot
be labled according to the rule.

RonL

thankn you very much now i understand those questions

but could anyone please explain c)?
That is the hardest question

thankyuo

Originally Posted by shosho

The numbers 1 to 25 are to be placed on a black and white 5x5 grid so that each number, except 1, is next to (horizontally or vertically), the number one less than itself. eg.



a) Explain why number 1 must be placed on a black square
b) Explain why the numbers 1, 3, 5, 7, 9 can not all be placed on one long diagonal
c) Show that in a completed numbering the sum of the numbers on a long diagonal must be at least 33 and that 33 can occur.

Any help is appreciated!
Thanks!

I don't have an answer to c) yet but the approach I will pursue (and you
might want to think about) is:

1. find a permitted numbering with a principle diagonal sum of 33.

2. The largest element on the diagonal if the diagonal sum is 32 or less
is 16. We know there is no numbering with a maximum of 11 on the diagonal.
Can we show that with a maximum of 12, 13, 14, 15 and 16 on the diagonal
we divide the board into two parts that a number path cannot cross. (that
is we try to extend the argument of part b) to fill in these cases)

RonL