194-187
Problems on Purchases
and Prices
cleverest daughter, thirty to the second, and fifty to the
third, saying:
'You should agree beforehand on the price at which
you'll sell the eggs and stick to it. All of you should
adhere to this price but I hope t
168-165
Brain-Twisting
Arrangements
and Permutations
The commander's tent is guarded by sentries housed
in eight other tents (Fig. 166). Initially in each of the
tents there were three sentries. Later the sentries were
allowed to visit each other and thei
170-165
Brain-Twisting
Arrangements
and Permutations
to see the result. Much to his dismay he found the
orchard almost devastated: instead of the 20 trees the
workman had left only 10 and cut 39.
"Why have you cut so many ? You were told to leave
20 trees
156-157
Optical
Illusions
Figure 143
Figure 144
Distance AB seems to be wider than distance AC.
which is equal to the former.
Figure 145
A
0
B
Holding Fig. 146 at eye level so that you glance
along it, you'll see the picture given on the right.
Close one
Brain-Twisting
Arrangements
and Permutations
two castles protected within the walls. The architect
objected that it was impossible to satisfy the condition
whilst the 10 castles had to be arranged four in each of
the five walls. But the prince insisted.
A
Optical
Illusions
Figure 146
paper. Shift the figure slightly sidewards and the pins
will swing.
Figure 147
Figure 149
Figure
148
If you view this figure for a long time, it'll seem to
you that the two cubes at the top and at the bottom
stand out alternat
Answers
A Clock Dial
As the sum of all the numbers on the face of the dial is 78, the sum of each of the six
sections mast be 78-f-6 = 13. This facilitates finding the solution that is shown in
Fig. 189.
Figure 189
J ^ T j Z ^ K .
Crescent
The answer is s
n
Answers
A Pond
It is possible to double the surface area of the pond with the square shape retained
and the oaks intact. The accompanying figure shows how this can be done. You can
Figure 197
easily see that the new area is twice the earlier, just draw
158-159
Figure
150
Optica!
Illusions
touching the walls from the inside, the box being open
at the bottom.
The intersections of the white lines in this figure seem
to have yellowish square spots that appear and
disappear, as if flashing. In actuality, the
Brain-Twisting
Arrangements
and Permutations
Figure 165
Jacob's garden
Peter's garden
Paul s
garden
Pranks of Guards
The following is an ancient problem having many
modifications. We'll discuss one of them.
Figure 166
162-163
Optical
Illusions
exceptionally simple cause. If we view the picture we
imagine the things shown in it, and it seems to us that
the thing has changed its position.
The same applies to the portraits. When we observe
a real face from the side, we se
Answers
In Six Rows
The requirement of the problem is easily met if the people are arranged in the form of
a hexagon as shown in the figure.
Figure 170
In Nine Squares
You don't touch the forbidden coin but shift the whole of the lower row upwards
(Fig. 1
Answers
vJ'
Navvies
It's easy to swallow the bait and think that if five navvies dug 5 metres of the ditch in
5 hours, then it would take 100 people to dig 100 metres in 100 hours. But that
argument is absolutely wrong, since the same five navvies would b
174-175
Answers
Peter's house
Figure
174
Jacob's garden
Peter's garden
Paul's garden
Pranks of Guards
The problem is easily solved by the following reasoning. For four guards to be able to
be absent unnoticed by the chief it's necessary that in rows / and
Answers
1Q ^
How Much are the Lemons?
We know that the 36 lemons cost as many roubles as they sell lemons for 16 roubles.
But 36 lemons cost
36 x (price of one lemon).
For 16 roubles one can have
16/(price of one lemon).
Hence,
36 x (price of one lemon) =
184-185
Problems with Squares
A Seamstress
A seamstress wants to cut out a piece of linen in the
form of a square. Having cut several pieces she checks
her work by bending each piece along its diagonal to
see if the edges coincide. If they do, she thinks,
Optical
Illusions
of contrast you can see that each strip is uniformly
darkened.
Figure 153
Look attentively for a minute at some point on this
"negative" portrait of Newton without moving your
eyes, then quickly shift your glance to a piece of white
pape
Answers
Figure 173
I
w
w
If*
I
T
' / i l i i i i M
Eight
Letters
The least number of moves is 23. These are as follows:
A B F E C A B F E C A B D H G A B D H G D E F
Squirrels and Rabbits
Shown below is the shortest way of the rearrangement The first numb
Problems with Squares
A Pond
There is a square pond (Fig. 194) with four old oaks
growing at its corners. It is required to expand the
Figure 194
pond so that its surface area be doubled, the square
shape being retained and the old oaks not destroyed or
s
Problems on Manual Work
Navvies
Five navvies excavate a 5-metre ditch in 5 hours. How
many navvies are required to dig 100 metres of ditch in
100 hours?
Lumberjacks
A lumberjack cuts a 5-metre log into 1-metre lengths. If
each cut takes 1.5 minutes, how l
Answers
Figure 176
four on each of the five lines. On the right of Fig. 176, four more solutions to the
problem are given.
An Orchard
The uncut trees were disposed as given in Fig. 177. These form five straight rows with
four trees in each.
Figure
ill
180-181
Answers
With Three Straight Lines
The problem is solved as follows:
Figure 186
Into Four Parts
The dash lines show the way in which the ground must be divided (Fig. 187).
Figure 187
To Make a Circle
The joiner has cut each of the boards into four
Answers
We can thus easily find the sum of the weights of No. 1, No. 2, No. 4, and No. 5:
110 kg + 121 kg = 231 kg. Subtracting this number from the total weight of the bags
(289 kg) gives the weight of No. 3, namely-58 kg.
Further, from the sum of No. 1
Skilful Cutting
and
Connecting
a square from five identical triangles like the ones you
have just used (the base is twice as long as the height).
You may cut one of the triangles into two parts but the
other four must be used as they are (Fig. 185b).
Figu
166-165
Brain-Twisting
Arrangements
and Permutations
How could they make it?
Observe the following rules:
(1) each animal may make several leaps at once;
(2) two animals may not seat on the same stump,
therefore they must only leap on a vacant stump.
Furt
176-177
Answers
The White
Mouse
The cat should first eat the mouse at which it is looking, i. e. the sixth one from the
white. Try it by beginning with this mouse and cross out every 13th mouse. You'll see
that the white mouse will be the last to be cross
186-187
Answers
Yet Another Parquet
Maker
The test might only show that the quadrilateral in question has right angles, i. e. that
it is a rectangle. But it fails to verify that all its sides are equal, as is seen in Fig. 200.
Figure 200
A Seamstress
The
160-161
Optical
Illusions
Figure 155
Figure 156
On the left you see a convex cross, on the r i g h t - a
concave one. But turn the book upside down and the
figures will change their places. Actually the figures are
identical, only they're shown at differe
190-187
This is the actual duration of work of the first peeler, the second one worked for
70 + 25 = 95 minutes.
In fact: 3 x 70 + 2 x 95 = 400.
Two
Workers
If each worker performs half the job individually, the first would need two days more
than the sec
192-193
Problems on Purchases
and Prices
How Much are the Lemons?
Three dozen lemons cost as many roubles as one can
have lemons for 16 roubles.
How much does a dozen lemons cost?
Raincoat, Hat and Overshoes
A raincoat, hat and overshoes are bought for 14