These solutions mark the desired spots
just barely within 1 cm of their target heights
by treating the lamps as
point sources
and by treating the book and table as
rectangular boxes.
As groundwork,
here is a quick inventory of readily available distances
that may be obtained through diagonals.
$\small\sqrt{ 60^2{+}\,80^2\strut}~ = ~ 100 ~ ~$cm
$\small\sqrt{ 130^2{-}\,90^2\strut}~\approx~ 93.8 ~$cm
$\small\sqrt{ 130^2{+}\,60^2\strut}~\approx~ 143.2 $ cm
$\small\sqrt{ 60^2{+}\,90^2\strut}~\approx~ 108.2 ~$cm
$\small\sqrt{ 130^2{-}\,80^2\strut}~\approx~ 102.5 ~$cm
$\small\sqrt{ 130^2{+}\,80^2\strut}~\approx~ 152.6 ~$cm
$\small\sqrt{ 80^2{+}\,90^2\strut}~\approx~ 120.4 ~$cm
$\small\sqrt{ 130^2{-}\,60^2\strut}~\approx~ 115.3 $ cm
$\small\sqrt{ 130^2{+}\,90^2\strut}~\approx~ 158.1 ~$cm
SPOT 1
a. Begin with the table in the room’s front right corner.
b. Place the book on the floor so that one corner of it
marks the table’s free corner,
which is now a
$\small \sqrt{60^2{+}\,90^2 \strut} = {\scriptsize\sim}108.2$ cm
diagonal away from the room’s corner.
c. Move the table out of the way.
d. Place one end of the stick on the floor
at the book’s marker corner
and lean the other end into the room’s corner.
The top of the stick now marks a height of
$\small \sqrt{130^2-108.2^2 \strut} = {\scriptsize\sim}72.1~$cm,
just within 1 cm of the target 73 cm.
Back story 1.
To form a right triangle
with a 130 cm hypotenuse
and a 73 cm leg,
the perfect distance from the corner
to the bottom end of the stick
would be
$\small \sqrt{130^2-73^2 \strut} = {\scriptsize\sim}107.6~$cm.
Only one diagonal from the groundwork list was close enough to try,
and it worked.
SPOT 2
a. Begin with the table’s 90 cm side
flush against the rear wall.
b. Slide the table along the wall
until the stick exactly reaches diagonally along the floor
from the room’s rear right corner
to the desk’s front right corner,
60 cm out from the rear wall.
c. Place the book on the floor so that it is flush with both
the rear wall and the table’s right wall.
The left edge of the book now marks a distance
$\small \sqrt{130^2-60^2 \strut} = {\scriptsize\sim}115.3 ~$cm
along the wall from the room’s rear right corner.
d. Move the table so that its 60 cm side
is flush with the right wall
and so that its 90 cm side casts no shadow on the floor
from the ceiling-centered overhead lamp.
e. Place the stick on the floor
along that no-shadow side of the table.
The stick now marks a line parallel to the rear wall
exactly 150 cm from the room’s rear right corner.
f. Slide the table leftward along the stick.
g. Visually align the table’s left side
with the book’s marker edge, leaving a gap of
$\small {\scriptsize\sim}115.3 - 90 = {\scriptsize\sim}25.3~$cm
between the right side of the desk and the wall.
h. Stand the stick along the front right vertical edge of the table.
i.
Adjust the desk lamp so that
its bulb is on the front left corner of the table top
and shines toward the top of the stick, which is
90 cm to the right and
50 cm higher.
j.
The top of the stick’s shadow now marks a height
of
$\small 130 + \dfrac{\raise-.5ex{{\scriptsize\sim}25.3}}{90}
{\scriptsize\times \,} 50
~ = ~ { \scriptsize\sim }144.1~$cm
above the floor,
just within 1 cm of the target 145 cm.
Back story 2.
Where can the target distance of 150 cm
from the back wall be found?
The ceiling lamp!,
which can also provide a perpendicular line from the right wall
thanks to the table’s perpendicular vertical sides.
Hoping to cast the stick’s shadow
145 cm up the wall,
the two simplest alignments along the desk’s top edges are
60 cm and
90 cm away from corners where the desk lamp might be positioned.
As the stick is
50 cm taller than the table,
and SPOT 2 is
15 cm higher than the stick’s top,
the perfect distances of the lamp from the wall would
be
$\small 60 + \dfrac{\raise-.5ex{60}}{50} {\scriptsize\times \,} 15
\, = \, 78~$cm
and
$\small 90 + \dfrac{\raise-.5ex{90}}{50} {\scriptsize\times \,} 15
\, = \, 117~$cm.
Again, only one diagonal from the groundwork list was close enough to try,
and it worked.
