What is the smallest number chart on which this tile can be placed on so its centre squares covers the number 41?
A number chart is a square grid with the smaller squares number consecutively 1,2,3 accross the top row from left to right, then the next row from left to right and so on . A number chart is named after its largest number Eg...
Question : Another tile is a 5 times 5 square of 25 small squares. What is the smallest number chart on which this tile can be placed on so its centre squares covers the number 41?
The smallest number chart that 41 appears in is a 7-chart. So we restrict our enquiry to 7 and larger.
Note that the center square of a 5-chart is in the 3rd row and 3rd column. Since we are only interested in 7-charts or larger, we do not have to worry about the row placement, so we concentrate on the columns.
In order to cover with the middle square of a 5-chart, 41 must be at least 3 columns away from an edge.
In a 7-chart, the number 41 occurs in the 6th column of the 6th row-- this is next to an edge so it will not work.
In an 8-chart, the number 41 is in the first column or on the edge so this will not work.
In a 9-chart, the number 41 occurs in the 5th column -- this works as there are 4 columns to the left, 4 columns to the right, 4 columns below, and 4 columns above.
I've linked to an article on modular arithmetic -- you will notice that the column placement of the numbers depends on the value of the number in question modulo the size of the chart. `41-= 5(mod9)` so it is in the 5th column. (This means that when you throw out multiples of 9, you are left with a remainder of 5)