Here's an easy puzzle, if you have all the facts to hand.

If you cut two A4 sheets of paper along the diagonal, you can arrange the
four triangular pieces into a square, with the pieces touching, but not overlapping and the diagonals forming the outer edge of the square. There
will be a small uncovered area in the centre of this arrangement.

What proportion of the area of the larger square does the central
uncovered square area form?

--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 28/04/2025 08:55, David Entwistle wrote:

Here's an easy puzzle, if you have all the facts to hand.

If you cut two A4 sheets of paper along the diagonal, you can arrange the four triangular pieces into a square, with the pieces touching, but not overlapping and the diagonals forming the outer edge of the square. There will be a small uncovered area in the centre of this arrangement.

What proportion of the area of the larger square does the central
uncovered square area form?

No doubt I've screwed up somewhere, but just in case I haven't
here's a spoiler alert:

.....Spoiler! Don't read down if you don't want to know.
.....Spoiler! Don't read down if you don't want to know
.....Spoiler! Don't read down if you don't want to kno
.....Spoiler! Don't read down if you don't want to kn
.....Spoiler! Don't read down if you don't want to k
.....Spoiler! Don't read down if you don't want to
.....Spoiler! Don't read down if you don't want to
.....Spoiler! Don't read down if you don't want t
.....Spoiler! Don't read down if you don't want
.....Spoiler! Don't read down if you don't want
.....Spoiler! Don't read down if you don't wan
.....Spoiler! Don't read down if you don't wa
.....Spoiler! Don't read down if you don't w
.....Spoiler! Don't read down if you don't
.....Spoiler! Don't read down if you don't
.....Spoiler! Don't read down if you don'
.....Spoiler! Don't read down if you don
.....Spoiler! Don't read down if you do
.....Spoiler! Don't read down if you d
.....Spoiler! Don't read down if you
.....Spoiler! Don't read down if you
.....Spoiler! Don't read down if yo
.....Spoiler! Don't read down if y
.....Spoiler! Don't read down if
.....Spoiler! Don't read down if
.....Spoiler! Don't read down i
.....Spoiler! Don't read down
.....Spoiler! Don't read down
.....Spoiler! Don't read dow
.....Spoiler! Don't read do
.....Spoiler! Don't read d
.....Spoiler! Don't read
.....Spoiler! Don't read
.....Spoiler! Don't rea
.....Spoiler! Don't re
.....Spoiler! Don't r
.....Spoiler! Don't
.....Spoiler! Don't
.....Spoiler! Don'
.....Spoiler! Don
.....Spoiler! Do
.....Spoiler! D
.....Spoiler!
.....Spoiler!
.....Spoiler
.....Spoile
.....Spoil
.....Spoi
.....Spo
.....Sp
.....S
.....
....
...
..
.



If this is true for A4, it is true for all A*. All A* sheets have
an aspect ratio of 1/sqrt(2). I will choose * so that the shorter
side has length 1.

Pythag tells us that the diagonals are sqrt(1^2 + sqrt(2)^2), or
sqrt(3).

The area of a square with sides sqrt(3) is 3.

The area of two sheets is 2 * sqrt(2), so we have a central
square of area 3-(2*sqrt(2)), and thus a ratio of:

3 - (2 * sqrt(2))
-----------------
3


--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Mon, 28 Apr 2025 09:26:21 +0100, Richard Heathfield wrote:

No doubt I've screwed up somewhere, but just in case I haven't here's a spoiler alert:

Looks good to me. Well done.

--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> writes:

Here's an easy puzzle, if you have all the facts to hand.

If you cut two A4 sheets of paper along the diagonal, you can arrange the four triangular pieces into a square, with the pieces touching, but not overlapping and the diagonals forming the outer edge of the square. There will be a small uncovered area in the centre of this arrangement.

What proportion of the area of the larger square does the central
uncovered square area form?

*looks up a4 paper dimensions. Since the imperial dimensions are stupid,
I'll use the listed 210x297mm

My solution below
My solution below
My solution below
My solution below
My solution below
My solution below
My solution below
My solution below
My solution below
My solution below
My solution below
My solution below
My solution below


Cutting a4 diagnolly creates two congruent triangles with a hypotenuse of

c^2 = 210^2 + 297^2 -> c^2 = 44100 + 88209 -> c^2 = 132309

thus

C is the length of each side of the squares. We don't need to square
root the side since we only need C^2 - the area of the square.

Area of the square is C^2 = 132309 which we already established this from the pythagorean theorem.

Then we calculate the area of one of the triangles, formula for triangle
area is 1/2(b*h) = 1/2(210*297) -> 1/2(31185) -> 31185

Multiple that by 4, since all the triangles are congruent = 124740

Subtract that from the square area. 132309 - 124740 = 7569mm

7569mm is the area of the blank center.

Looks like someone had an extra credit problem for homework.

Daniel

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Wed, 28 May 2025 12:42:24 -0700, Daniel wrote:

Looks like someone had an extra credit problem for homework.

:) I wish. Homework is a dim and distant memory for me.

--
David Entwistle

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 28/04/2025 08:55, David Entwistle wrote:

Here's an easy puzzle, if you have all the facts to hand.

If you cut two A4 sheets of paper along the diagonal, you can arrange the four triangular pieces into a square, with the pieces touching, but not overlapping and the diagonals forming the outer edge of the square. There will be a small uncovered area in the centre of this arrangement.

What proportion of the area of the larger square does the central
uncovered square area form?

I handed this to ChatGPT, and it came up with the right numerical
(and so approximate) answer straight away, but I had to ask it to
generalise before it uncovered the exact result.

--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)