David Williams
@bewilda.bsky.social
Retired Mathematics/Chemistry teacher. Shrewsbury Town fan.
Septuagenarian.
Septuagenarian.
127 is the first Friedman prime. My grandson and I were watching Australian Open tennis singles matches today and he asked how many players are in the main draw. My response was 128 and there has to be 127 matches to decide the winner (127 players have to lose). Tennis is so boring.
January 21, 2026 at 10:26 AM
127 is the first Friedman prime. My grandson and I were watching Australian Open tennis singles matches today and he asked how many players are in the main draw. My response was 128 and there has to be 127 matches to decide the winner (127 players have to lose). Tennis is so boring.
Find the positive integer A such that exactly two of the following statements are true:
(a) A + 82 is the square of an integer; (b) the last digit of A is 5;
(c) A − 7 is the square of an integer.
(a) A + 82 is the square of an integer; (b) the last digit of A is 5;
(c) A − 7 is the square of an integer.
June 16, 2025 at 7:14 PM
Find the positive integer A such that exactly two of the following statements are true:
(a) A + 82 is the square of an integer; (b) the last digit of A is 5;
(c) A − 7 is the square of an integer.
(a) A + 82 is the square of an integer; (b) the last digit of A is 5;
(c) A − 7 is the square of an integer.
Consider the triangle ABC with AB = n^2, BC = 2n + 1
and AC = n(n+ 1) for all n > 1.
Use the cosine rule and the sine rule to find angles B and C for at least 3 integer values of n.
What do you notice? Can you prove your conjecture?
What ”sublime” triangle is formed when AC = BC?
and AC = n(n+ 1) for all n > 1.
Use the cosine rule and the sine rule to find angles B and C for at least 3 integer values of n.
What do you notice? Can you prove your conjecture?
What ”sublime” triangle is formed when AC = BC?
April 24, 2025 at 8:09 PM
Consider the triangle ABC with AB = n^2, BC = 2n + 1
and AC = n(n+ 1) for all n > 1.
Use the cosine rule and the sine rule to find angles B and C for at least 3 integer values of n.
What do you notice? Can you prove your conjecture?
What ”sublime” triangle is formed when AC = BC?
and AC = n(n+ 1) for all n > 1.
Use the cosine rule and the sine rule to find angles B and C for at least 3 integer values of n.
What do you notice? Can you prove your conjecture?
What ”sublime” triangle is formed when AC = BC?
Is this a valid approach?
∫ln(x - 4)dx = (x - 4)ln(x - 4) - ∫(x - 4)/(x - 4)dx
= (x - 4)ln(x - 4) - ∫dx
= (x - 4)ln(x - 4) - x + c
∫ln(x - 4)dx = (x - 4)ln(x - 4) - ∫(x - 4)/(x - 4)dx
= (x - 4)ln(x - 4) - ∫dx
= (x - 4)ln(x - 4) - x + c
March 30, 2025 at 7:22 PM
Is this a valid approach?
∫ln(x - 4)dx = (x - 4)ln(x - 4) - ∫(x - 4)/(x - 4)dx
= (x - 4)ln(x - 4) - ∫dx
= (x - 4)ln(x - 4) - x + c
∫ln(x - 4)dx = (x - 4)ln(x - 4) - ∫(x - 4)/(x - 4)dx
= (x - 4)ln(x - 4) - ∫dx
= (x - 4)ln(x - 4) - x + c
It's the 43rd day of the year and it's a stifling 43⁰ C today in the city where I live. Fun fact: -40⁰ C = -40⁰ F.
February 12, 2025 at 9:36 AM
It's the 43rd day of the year and it's a stifling 43⁰ C today in the city where I live. Fun fact: -40⁰ C = -40⁰ F.
Fun fact: Today is the 42nd day of the year (Feb. 11th).
10! seconds is exactly 42 days.
10! seconds is exactly 42 days.
February 11, 2025 at 5:13 PM
Fun fact: Today is the 42nd day of the year (Feb. 11th).
10! seconds is exactly 42 days.
10! seconds is exactly 42 days.
Along the lines of my previous post.
Find the values of k such that the curves
y = e^(-2x) and y = e^(-x) + k
intersect at two points.
Find the values of k such that the curves
y = e^(-2x) and y = e^(-x) + k
intersect at two points.
January 18, 2025 at 1:45 AM
Along the lines of my previous post.
Find the values of k such that the curves
y = e^(-2x) and y = e^(-x) + k
intersect at two points.
Find the values of k such that the curves
y = e^(-2x) and y = e^(-x) + k
intersect at two points.
From an NZQA exam.
Find the value(s) of p for which the equation
x -2√(x + p) = -5
has one real solution.
Find the value(s) of p for which the equation
x -2√(x + p) = -5
has one real solution.
January 17, 2025 at 11:28 AM
From an NZQA exam.
Find the value(s) of p for which the equation
x -2√(x + p) = -5
has one real solution.
Find the value(s) of p for which the equation
x -2√(x + p) = -5
has one real solution.
Show that the area under the graph of y = x²/(x² + 4) between x = -2 and x = 2 is equal to the shade area between the square and the circle.
www.desmos.com/calculator/l...
www.desmos.com/calculator/l...
arctan
www.desmos.com
January 10, 2025 at 11:45 AM
Show that the area under the graph of y = x²/(x² + 4) between x = -2 and x = 2 is equal to the shade area between the square and the circle.
www.desmos.com/calculator/l...
www.desmos.com/calculator/l...
An interesting alternative to IBP where functions are proportional to their second derivatives.
www.johndcook.com/blog/2023/01...
www.johndcook.com/blog/2023/01...
Avoid having to integrate by parts twice
Integrating the product of two functions like exp(ax), sin(bx), cosh(cx), etc. without having to integrate by parts.
www.johndcook.com
December 12, 2024 at 9:29 AM
An interesting alternative to IBP where functions are proportional to their second derivatives.
www.johndcook.com/blog/2023/01...
www.johndcook.com/blog/2023/01...
The sides of a triangle are in an arithmetic progression with a common difference of 6. The largest angle is 120°.
Find the exact area of the triangle (measurements in cm).
Find the exact area of the triangle (measurements in cm).
December 11, 2024 at 5:41 PM
The sides of a triangle are in an arithmetic progression with a common difference of 6. The largest angle is 120°.
Find the exact area of the triangle (measurements in cm).
Find the exact area of the triangle (measurements in cm).
Love the numberplate.
December 7, 2024 at 2:07 PM
Love the numberplate.
Anyone 48 on 6/12/24?
December 5, 2024 at 7:28 PM
Anyone 48 on 6/12/24?
Investigating students' "algebraic sense" - a sixth sense perhaps.
Consider the function f(x)=(x^2+x+k)/(x^2-1)
For what values of k will f(x) have no stationary points?
Consider the function f(x)=(x^2+x+k)/(x^2-1)
For what values of k will f(x) have no stationary points?
December 4, 2024 at 2:43 PM
Investigating students' "algebraic sense" - a sixth sense perhaps.
Consider the function f(x)=(x^2+x+k)/(x^2-1)
For what values of k will f(x) have no stationary points?
Consider the function f(x)=(x^2+x+k)/(x^2-1)
For what values of k will f(x) have no stationary points?
From a final year of secondary school mathematics exam in Victoria today:
For what values of b (a real constant) does the graph of the function
g(x) = (x⁴ + b)/(1 - x²)
have one, three or five stationary points?
For what values of b (a real constant) does the graph of the function
g(x) = (x⁴ + b)/(1 - x²)
have one, three or five stationary points?
November 13, 2024 at 10:48 AM
From a final year of secondary school mathematics exam in Victoria today:
For what values of b (a real constant) does the graph of the function
g(x) = (x⁴ + b)/(1 - x²)
have one, three or five stationary points?
For what values of b (a real constant) does the graph of the function
g(x) = (x⁴ + b)/(1 - x²)
have one, three or five stationary points?
What is the measure of the largest angle of the triangle with sides, 3319, 1469 and 2331 units?
November 7, 2024 at 7:46 PM
What is the measure of the largest angle of the triangle with sides, 3319, 1469 and 2331 units?
What do you notice?
The function, f(x) = √(7 - 3x) and its inverse.
The function, f(x) = √(7 - 3x) and its inverse.
October 15, 2024 at 9:01 AM
What do you notice?
The function, f(x) = √(7 - 3x) and its inverse.
The function, f(x) = √(7 - 3x) and its inverse.