Best Approximations
Let $x$ be a real number. A best approximation to $x$ for the denominator bound $d$ is a rational number $\frac r s $ in reduced form,...
-
John Rhodes
-
August 15, 2023
Prize Strings
A particular school offers cash rewards to children with good attendance and punctuality. If they are absent for three consecutive days or late on more than one...
-
John Rhodes
-
August 15, 2023
Maximising a Weighted Product
Let $S_m = (x_1, x_2, \dots , x_m)$ be the $m$-tuple of positive real numbers with $x_1 + x_2 + \cdots + x_m = m$ for which...
-
John Rhodes
-
August 15, 2023
Hyperexponentiation
The hyperexponentiation or tetration of a number $a$ by a positive integer $b$, denoted by $a\mathbin{\uparrow \uparrow}b$ or $^b a$, is recursively defined by: $a \mathbin{\uparrow \uparrow}...
-
John Rhodes
-
August 15, 2023
Semiprimes
A composite is a number containing at least two prime factors. For example, $15 = 3 \times 5$; $9 = 3 \times 3$; $12 = 2 \times...
-
John Rhodes
-
August 15, 2023
Connectedness of a Network
Here are the records from a busy telephone system with one million users: RecNrCallerCalled $1$$200007$$100053$$2$$600183$$500439$$3$$600863$$701497$$\cdots$$\cdots$$\cdots$ The telephone number of the caller and the called number in record...
-
John Rhodes
-
August 15, 2023
Number Mind
The game Number Mind is a variant of the well known game Master Mind. Instead of coloured pegs, you have to guess a secret sequence of digits....
-
John Rhodes
-
August 15, 2023
Triangles Containing the Origin
Consider the set $I_r$ of points $(x,y)$ with integer co-ordinates in the interior of the circle with radius $r$, centered at the origin, i.e. $x^2 + y^2...