Place four pebbles on the sand in the form of a square. Keep adding
as few pebbles as necessary to double the area. How many extra
pebbles are added each time?
The nth term of a sequence is given by the formula n^3 + 11n . Find
the first four terms of the sequence given by this formula and the
first term of the sequence which is bigger than one million. Prove
that all terms of the sequence are divisible by 6.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Alexander, from Wilson's School, wrote:
Phoebe first drew one “down,” and then as many “inverted Cs” as needed to get seven squares. Therefore:
Alice first drew seven “alongs” at the top, and then seven “alongs” at the bottom. She then used eight “downs” to connect the gaps in between two parallel “alongs.” Therefore:
Luke first drew one “square,” and then six “inverted Cs” to make seven squares. Therefore:
Great! Laeticia, from Woodbridge High School, also gave correct general formulas here. She went on to comment on one of our later puzzles:
If the height is h and the width is w, then the perimeter is 2h + 2w. Then there are (h+1)(w+1) dots, and w(h+1) + h(w+1) lines.
Niharika solved the rest of our questions:
Each L-shape has an 'inner' L-shaped line and an 'outer' L-shaped line. If an L-shape has height and width n, then the outer L has length 2n and the inner L has length 2(n-1). There are 2 lines remaining on the ends of the L, so altogether the perimeter is 4n.
There are 2(n-1) lines left inside the L-shape, so the shape is made of 6n - 2 lines.
The number of squares in an L-shape is 2n-1: think of each L as two rectangles of height 1 and width n that overlap in one square.
Think of these as two separate overlapping squares, each with $n^2$ dots. They overlap in one dot, so in total there are $2n^2 - 1$ dots, and so $2n^2 - n - 1$ white dots.
In each square there are 2n(n-1) lines, and the lines never overlap, so in total there are 4n(n-1) lines.
Square of squares:
We can split a pattern like this of side length n up into four rectangles of height 1 and length n-1, so there are 4(n-1) edge squares.
There are 4n lines making up the outer square and 4(n-2) lines making up the inner square. There are 4(n-1) lines left in the middle. This gives 12(n-1) lines in total.
Dots and more dots:
Inside the squares there are $n^2$ dots, and on the vertices there are $(n+1)^2$ dots, so in total there are $2n^2 + 2n + 1$ dots.
In each row there are n lines, and there are n+1 rows, so the rows contribute n(n+1) lines. Similarly the columns contribute n(n+1) lines, so together there are 2n(n+1) lines.
Rectangle of dots:
There are 4n horizontal lines and 3n vertical lines, so 7n lines in total.
On each row there are 2n+1 dots, and there are n+1 rows, so there are (2n+1)(n+1) dots.
Fantastic! Thank you all.