# From the people behind the Alan Turing Cryptography Competition.

# Puzzle 3

Given two sets $A$ and $B$ of numbers, Alex wants to understand the set $A+B$ made by adding together all combinations of a number from the set $A$ and a number from the set $B$. For example, $A=\{1,2,7\}$ and $B=\{3,5\}$ then $$A+B=\{1+3,1+5,2+3,2+5,7+3,7+5\}=\{4,6,5,7,10,12\}$$To do this Alex first makes a list of all points with coordinates $(x,y)$, where $x$ is in $A$ and $y$ is in $B$. He then replaces each $(x,y)$ with the closest point $(z,z)$ on the straight line through $(0,0)$ at angle 45 degrees to the horizontal. Finally he multiplies the distance from $(0,0)$ to $(z,z)$ by $\sqrt{2}$ to make a number $r=x+y$.

For example, $(2,5)$ is replaced by $(z,z)=(3.5,3.5)$ which is at distance $\frac{7}{\sqrt{2}}$ from $(0,0)$, so we end up with the final number $r=7$.

The set $4A+7B$ consists of all points $4a+7b$ where $a$ is in $A$ and $b$ is in $B$. This can be constructed by a similar method, except rather than mapping $(x,y)$ to the closest point on the line at angle $45$ degrees to the horizontal a different angle between 0 and 90 degrees is used, and rather than multiplying by $\sqrt{2}$ at the end we multiply by a different number.

Which angle between $0$ and $90$ degrees should be used to construct $4A+7B$? Write down your angle in degrees to 3dp.

This problem was first solved on Wed 10th February at 4:01:45pm
Mathsbombe Competition 2021 is organised by the The Department of Mathematics at The University of Manchester.
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