On a plane there is a square, and invisible ink is dotted at a point P. A person with special glasses can see the spot. If we draw a straight line, then the person will answer the question of on which side of the line does P lie $($ if P lies on the line, then he says that P lies on the line $)$.
What is the smallest number of such questions you need to ask to find out if the point P is inside the square?
During the ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced with during the previous dance, or more intelligent, but most of the men $($ at least 80% $)$ – with a girl who was at the same time more beautiful and more intelligent. Could this happen? $($ There was an equal number of boys and girls at the ball.$)$
Decipher the following rebus $\\$
$\\$
All the digits indicated by the letter “E” are even $($ not necessarily equal $)$; all the numbers indicated by the letter O are odd $($ also not necessarily equal $)$.
For the convenience of further reasoning, we replace all even numbers with vowels, and odd ones with consonants, bearing in mind that the same number can correspond to different letters. Therefore the rebus will take the following form:$\\$
$\\$
Note that the numbers AEB and YHK differ by the second digit, so C $>$ 1. For C $>$ 3 or A $>$ 2, the product $AEB \times C$ will be four-digit, therefore C = 3, A = 2. Hence, I is at most 2, that is it is equal to 2. It follows that D = 9 $($ for a smaller value of D, the expression $AEB \times D$ will be less than 2100, and IFUG $>$ 2100 $)$ Y = 8 $($ otherwise the entire product will not be a five-digit number $)$. E = 8 $($ if E $<$ 8, then YHK $<$ 810 $)$. If B varies from 1 to 9, then the number represented by IFUG will vary from 2529 to 2601, therefore F = 5, and B $<$ 9. H = 5 $($ with B changing from 1 to 7, YHK will vary from 843 to 861 $)$. K = 5 $($ otherwise the number 85K will not be divisible by 3 $)$.
285 × 39 = 11115.
In Neverland, there are magic laws of nature, one of which reads: “A magic carpet will fly only when it has a rectangular shape.” Frosty the Snowman had a magic carpet measuring $9 \times 12$. One day, the Grinch crept up and cut off a small rug of size $1 \times 8$ from this carpet. Frosty was very upset and wanted to cut off another $1 \times 4$ piece to make a rectangle of $8 \times 12$, but the Wise Owl suggested that he act differently. Instead he cut the carpet into three parts, of which a square magic carpet with a size of $10 \times 10$ could be sown with magic threads. Can you guess how the Wise Owl restructured the ruined carpet?
Think about how the magic carpet looked after the Grinch cut off a piece of it.
After the Grinch messed up the carpet, Frosty could cut off a piece of size $1 \times 4$ from this carpet and turn it into a carpet measuring $8 \times 12$. This means that after the actions of the Grinch the carpet looked as shown in Fig. 1. The Wise Owl cut this carpet as shown in Fig. 2, and sewed it as shown in Fig. 3.
See the picture on the right.
What is the maximum number of pieces that a round pancake can be divided into with three straight cuts?
Remember task 11.
Since we are cutting a cylindrical cake and not a flat tart we should think of cuts in various planes. On the picture shown the way to cut the cake into 8 pieces with 2 vertical cuts and one horizontal.
8 pieces.
How can you divide a pancake with three straight sections into 4, 5, 6, 7 parts?
The number of parts depends on whether the cuts cross each other inside the pancake.
We draw in the pancake three straight lines and consider the points of their intersection. Depending on where these points are located, a specific number of parts will be obtained. To get 4 pieces, you need to place all three points outside the pancake $($Figure 1$)$. The transfer of one of these points from outside of the border of the pancake to the inside adds one part. So, to get 5 pieces, you have to transfer one point to the inside of the pancake $($Figure 2$)$, 6 – one more point needs to be moved inside the pancake $($Figure 3$)$, 7 – place all the three intersection points inside the pancake $($Figure 4$)$.
See the figure.
Two people had two square cakes. Each person made 2 straight cuts from edge to edge on their cake. After doing this, one person ended up with three pieces, and the other with four. How could this be?
