Santa's score draw

Santa has four houses to go and notices he has 20 identical stocking fillers in his sleigh, in addition to the presents the children in those houses have asked for.

In how many distinct ways can he distribute these between the four houses if each house has to get at least one?

Fault-free tilings

A *fault-free tiling* is an arrangement of tiles in an *m*×*n* grid such that there are no vertical or horizontal lines that can be placed on the grid without crossing one of the tiles.

In the image above a 6×4 and a 6×8 grid have been tiled with 2×1 dominoes. The 6×4 tiling has 2 vertical fault lines and is therefore not fault free. The 6×8 tiling is fault-free.

Is it possible, using 2×1 dominoes, to find fault-free tilings of:

- a 6×5 grid
- a 6×6 grid
- a 6×7 grid

Ritangle 2020

A rectangle has sides of length 12 and 8 units. A square of side *c* is drawn in one corner, creating the rectangular areas *P*, *Q*, *R* and *S* as in the diagram. What is the minimum value that (*Q* + *R*)/(*P* + *S*) can take?

This problem was taken from Ritangle 2017. Ritangle is a competition for teams of students aged 16 - 18 of A level Mathematics, the IB and Scottish Highers. Registration for Ritangle 2020 starts opens on 5th October 2020: integralmaths.org/ritangle.

Path of the midpoint

M is the midpoint of the points of intersection of *y*=1/*x* and *y*=2*x*+*c*. What path does M trace as *c* varies?

Function of a function

The function f(*x*)=(*x*+1)/(*x*+2).

Find f^{2}(*x*), f^{3}(*x*), f^{4}(*x*)..., where f^{2}(*x*)=f(f(*x*)).

Area of an arbelos

An arbelos is the shape created inside a semi-circle with diameter AB when two further semi-circles are drawn to a point C on AB so that their diameters are AC and BC. The arbelos is the area inside the larger semi-circle but outside the two smaller semi-circles: shown as the blue region below.

Show that the area of an arbelos is the same as the area of a circle with diameter CD where D is the point on the larger semi-circle directly above C: shown as the red circle in the picture above.

Walk on by

Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on that day?

9 dots, 10 circles

In the 3×3 square grid of dots shown the dots are spaced one unit apart (vertically and horizontal).

Construct circles with radius √1, √2, √3, √4, √5, √6, √7, √8, √9 and √10 with a pair of compasses using just these points and any intersection points of circles created.

You can try an interactive version of this problem at: geogebra.org/m/wfegkq6f

This is adapted from a problem by Ed Southall: twitter.com/edsouthall

Nowt taken out

What is the sum of all the three-digit numbers that don't include a zero? For example 242 is included in the sum but 202 and 22 are not.

What is the sum for four-digit numbers? What is the sum for *n*-digit numbers?

Dotty squares

‘Dotty squares’ are squares whose vertices are on the dots of a grid spaced 1 unit apart. The two dotty squares shown have areas 4 and 10.

Show that it is not possible to draw a dotty square whose area is 3, 7, 11, ... or any number of the form 4*n*+3 (where *n* is an integer).

Autobiographical numbers

An autobiographical number is a number whose digits describe itself starting with the first digit giving the number of zeros, the next digit giving the number of ones, and so on. 2020 is an autobiographical number as it has 2 zeros, 0 ones, 2 twos and 0 threes.

What other numbers are autobiographical?