EntranceInteger and fractional parts of a number. Integer and fractional parts of the number III
Mathematical games and entertainment
Favorites
Editor Kopylova A.N.
Tech. Editor Murashova N.Ya.
Proofreader Secheiko L.O.
Delivered for recruitment on September 26, 2003. Signed for publication on December 14, 2003. Format 34×103¼. Phys. oven l. 8.375. Conditional oven l. 13.74. Uch. ed. l. 12.88. Circulation 200,000 copies. Order No. 279. Book price 50 rub.
Domoryad A.P.
Mathematical games and entertainment. Favorites. – Volgograd: VSPU, 2003, - 20 p.
The book presents selected problems from the monograph by Domoryad A.P. “Mathematical Games and Entertainment”, which was published in 1961 by the State Publishing House of Physical and Mathematical Literature in Moscow.
ISBN 5-09-001292-X BBK 22.1я2я72
©VGPU Publishing House, 2003
Determining the intended number using three tables
Spread numbers from 1 to 60 in a row in each of the three tables so that in the first table they stand in three columns of twenty numbers each, in the second - in four columns of 15 numbers each, and in the third - in five columns of 12 numbers each each (see Fig. 1), it is easy to quickly determine the number N (N≤) conceived by someone if the numbers α, β, γ of the columns containing the conceived number in the 1st, 2nd and 3rd are indicated in the tables: N will be equal to the remainder of dividing the number 40α+45β+36γ by 60 or the sum (40α+45β+36γ) modulo 60. For example, with α=3, β=2, γ=1:
40α+45β+36γ=0+30+36=6(mod60), i.e. N=6
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Fig.1
A similar question may arise for numbers up to 420, placed in four tables with three, four, five and seven columns: if α, β, γ are the numbers of the columns in which the intended number appears, then it is equal to the remainder of the division of the number 280α+ 105β+336+120δ by 420.
Tapeworm
A game called tapeworm is played on a board with thirty-three squares.
Such a board can be easily obtained by covering the chessboard with a sheet of cardboard with a cross-shaped cutout.
In the figure, each cell is indicated by a pair of numbers indicating the numbers of the horizontal and vertical rows at the intersection of which the cell is located. At the beginning of the game, all cells, with the exception of one, are occupied by checkers.
It is required to remove 31 checkers, and an empty “starting” cell is specified ( a,b) and “final” ( c,d), on which the checker that survived at the end of the game should be placed. The rules of the game are
are: any checker can be removed from the board if next to it (in the horizontal or vertical direction) there is a checker on one side (“removing”), and on the opposite side there is an empty square on which the “removing” The checker must be transferred at the same time.
From game theory it follows that there will be a solution if and only if a c(mod3) and b d(mod3).
Let us give an example of a problem in which cell (44) is both the initial and final cell.
- 64-44
- 56-54
- 44-64
- 52-54
- 73-53
- 75-73
- 43-63
- 73-53
- 54-52
- 35-55
- 65-45
- 15-35
- 45-25
- 37-35
- 57-37
- 34-36
- 37-35
- 25-45
- 46-44
- 23-43
- 31-33
- 43-23
- 51-31
- 52-32
- 31-33
- 14-34
- 34-32
- 13-33
- 32-34
- 34-54
- 64-44
Here, in the record of each move, the numbers of the original checker are indicated for the “removing” checker
Cells and the number of the cell on which it is placed (in this case, a checker is removed from the board,
standing on an intermediate square)
Try to remove 31 checkers:
a) Initial cell (5,7) and final cell (2,4);
b) Starting cell (5,5) and ending cell (5,2).
Addition and subtraction instead of multiplication
Before the invention of logarithm tables, to facilitate the multiplication of multi-digit numbers, so-called prostaspheric tables (from the Greek words “aphairesis” - taking away), which are tables of function values
For natural values of Z. Since for a and b integers (the numbers a+b and a-b are either both fair or both odd; in the latter case, the fractional parts of y and are identical), then multiplying a by b reduces the definition of a+b and a-b and, finally, the differences of numbers 
,taken tables.
To multiply three numbers you can use the identity
from which it follows that if you have a table of function values, calculating the product abc can be reduced to determining the numbers a+b+c, a+b-c, a+c-b, b+c-a and remember - using the table - the right side of the equality (*).
Let us give as an example such a table for .
The table shows: large numbers – values and small numbers – meaning k, where at
| UNITS | |||||||||||
| TENS | 1 3 | 2 16 | 5 5 | 9 0 | 14 7 | 21 8 | 30 9 | ||||
| 55 11 | 72 0 | 91 13 | 114 8 | 140 15 | 170 16 | 204 17 | 243 0 | 285 19 | |||
| 333 8 | 385 21 | 443 16 | 506 23 | 576 0 | 651 1 | 732 8 | 820 3 | 914 16 | 1016 5 |
It is not difficult, using the formula (*) and the table, to obtain:
9·9·9=820 3 – 30 9 – 30 9 – 30 9 =297,
17 8 4 = 1016 5 –385 21 – 91 13 + 5 5 = 544 (Check!!)
Function [x] (integer part of x)
The function [x] is equal to the largest integer not exceeding x (x is any real number). For example:
| |
Figure 2 shows a graph of this function, and the left end of each of the horizontal segments belongs to the graph (bold dots), and the right end does not.
of the diagonals of a square are equal to the same number 
If only the sums of numbers in any horizontal and vertical are the same, then the square is called semi-magical.
The magic 4-square is named after Dürer, a mathematician and artist of the 16th century, who depicted the square in the famous painting “Melancholy”.
By the way, the two lower middle numbers of this square form the number 1514 - the date of creation of the painting.
There are eight nine-cell magic squares. Two of them, which are mirror images of each other, are shown in the figure; the remaining six can be obtained from these squares by rotating them around the center by 90,180,270.
P1. The integer part of number.
Definition10. The integer part of a number is the largest integer r not exceeding.
It is denoted by the symbol or (less commonly (from the French “entire” - integer). If x belongs to the interval where r is an integer, then, that is, it is in the interval. Then, according to the properties of numerical inequalities, the difference will be in the interval. The number is shown as the fractional part of the number and denote Therefore, the fractional part of a number is always non-negative and does not exceed one, while the integer part of a number can take on both positive and non-positive values. Thus, and therefore
Properties:
- 1. arbitrary number;
- 2. when
For example:
The integer part function has the form
1. The function makes sense for all values of the variable x, which follows from the definition of the integer part of a number and the properties of numerical sets (the continuity of the set of real numbers, the discreteness of the set of integers and the infinity of both sets). Consequently, its domain of definition is the entire set of real numbers. .
- 2. The function is neither even nor odd. The domain of definition of the function is symmetrical with respect to the origin, but if then i.e. neither the parity condition nor the odd parity condition is satisfied.
- 3. The function y=[x] is not periodic.
4. The set of function values is a set of integers (by definition, the integer part of a number.
5. The function is unlimited, since the set of function values are all integers, the set of integers is unlimited.
6. The function is discontinuous. All integer values are discontinuity points of the first kind with a final jump equal to one. At each discontinuity point there is continuity on the right.
7. The function takes the value 0 for all belonging to the interval, which follows from the definition of the integer part of a number. Therefore, all values of this interval will be zeros of the function.
- 8. Taking into account the property of the integer part of a number, the function takes negative values for values less than zero, and positive values for values greater than one.
- 9. The function is piecewise constant and non-decreasing.
- 10. The function does not have extremum points, since it does not change the nature of monotonicity.
- 11. Since the function is constant on each interval, it does not take the largest and smallest values in the domain of definition
- 12. Graph of a function.
P2. Fractional part of a number
Properties:
1. Equality
The fractional part of a number function has the form
- 1. The function makes sense for the values of the variable x, which follows from the definition of the fractional part of a number. Thus, the domain of this function is all real numbers.
- 2. The function is neither even nor odd. The domain of definition of the function is symmetrical with respect to the origin of coordinates, but the parity condition and oddity condition are not satisfied
- 3. The function is periodic with the smallest positive period.
4. The function takes values on the interval, which follows from the definition of the fractional part of a number, i.e.
5. From the previous property it follows that the function is limited
6. The function is continuous on every interval, where is an integer, at every point the function suffers a discontinuity of the first kind. The jump is equal to one.
- 7. The function goes to zero for all integer values, which follows from the definition of the function, that is, all integer values of the argument will be zeros of the function.
- 8. The function takes only positive values throughout its entire domain of definition.
- 9. A function that strictly monotonically increases on each interval where n is an integer.
- 10. The function does not have extremum points, since it does not change the nature of monotonicity
- 11. Taking into account properties 6 and 9, on each interval the function takes a minimum value at point n.
12. Graph of a function.
Shkolnik Publishing House
Volgograd, 2003
A.P.Domoryad
BBK 22.1я2я72
Domoryad Alexander Petrovich
Mathematical games and entertainment
Favorites
Editor Kopylova A.N.
Tech. editor Murashova N.Ya.
Proofreader Secheiko L.O.
Delivered for recruitment on September 26, 2003. Signed for publication on December 14, 2003. Format 84x 108 ¼.Phys.print.l. 8.375. Conditional oven 13.74. Academician-ed.l. 12.82. Circulation 200,000 copies. Order No. 979. The price of the book is 50 rubles.
Domoryad A.P.
Mathematical games and entertainment: Favorites. - Volgograd: VSPU, 2003. - 20 p.
The book presents selected problems from the monograph by Domoryad A.P. “Mathematical Games and Entertainment”, which was published in 1961 by the state publishing house of physical and mathematical literature in Moscow.
ISBN5-09-001292-Х BBK22.1я2я72
© Publishing house "VGPU", 2003
Preface 6
Determining the intended number using three tables 7
Solitaire 8
Adding and subtracting instead of multiplying 11
Function [x] (integer part of x) 12
Figures from square pieces 14
Magic squares 16
Appendix 17
Preface
From the diverse material united by various authors under the general name of mathematical games and entertainment, several groups of “classical entertainment” can be distinguished, which have long attracted the attention of mathematicians:
Entertainment related to the search for original solutions to problems that allow for an almost inexhaustible variety of solutions; Usually they are interested in establishing the number of solutions, developing methods that give large groups of solutions or solutions that satisfy some special requirements.
Mathematical games, i.e. games in which two “moves” playing side by side, made alternately in accordance with the specified rules, strive towards a certain goal, and it turns out to be possible for any initial position to predetermine the winner and indicate how - with any moves of the opponent - he can achieve victory.
"Games of one person", i.e. entertainment in which, through a series of operations performed by one player in accordance with these rules, it is necessary to achieve a certain, pre-specified goal; here they are interested in the conditions under which the goal can be achieved, and are looking for the smallest number of moves necessary to achieve it.
Everyone can try, by showing persistence and ingenuity, to get interesting (their own!) results.
If such classical entertainment as, for example, composing “magic squares” may appeal to a relatively narrow circle of people, then composing, for example, symmetrical figures from the details of a cut square, searching for numerical curiosities, etc., without requiring any mathematical training, can bring pleasure to both amateurs and non-lovers of mathematics. The same can be said about entertainment that requires preparation in the 9-11 grades of high school.
Many entertainments and even individual problems can suggest topics for independent research for mathematics lovers.
In general, the book is intended for readers with a mathematical background in grades 10-11, although most of the material is accessible to ninth graders, and some questions are even accessible to students in grades 5-8.
Many paragraphs can be used by mathematics teachers to organize extracurricular activities.
Different categories of readers can use this book in different ways: people who are not keen on mathematics can get acquainted with the curious properties of numbers, figures, etc., without delving into the rationale for games and entertainment, taking individual statements on faith; We advise mathematics lovers to study individual parts of the book with pencil and paper, solving the proposed problems and answering individual questions proposed for reflection.
Determining the intended number using three tables
By placing numbers from 1 to 60 in a row in each of three tables so that in the first table they are in three columns of twenty numbers each, in the second - in four columns of 15 numbers each, and in the third - five columns of 12 numbers each each (see Fig. 1), it is easy to quickly determine the number N (N≤60) conceived by someone if the numbers α, β, γ of the columns containing the conceived number in the 1st, 2nd and 3rd are indicated tables: N will be exactly the remainder of dividing the number 40α+45β+36γ by 60 or, in other words, N will be exactly the smaller positive number comparable to the sum (40α+45β+36γ) modulo 60. For example, with α=3, β =2, γ=1:40α+45β+36γ≡0+30+36≡6 (mod60), i.e. N=6.
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A similar question can be solved for numbers up to 420, placed in four tables with three, four, five and seven columns: if - the numbers of the columns in which the intended number is, then it is equal to the remainder after dividing the number 280α+105β+336γ+120δ at 420.
Tapeworm
| 737773 | 747774 | 757775 | ||||
| 636663 | 642264 | 656665 | ||||
| 515551 | 555252 | 535553 | 544554 | 554455 | 555556 | 555557 |
| 414441 | 424442 | 434443 | 444444 | 454445 | 464446 | 474447 |
| 313331 | 323332 | 333333 | 343334 | 353335 | 363336 | 373337 |
| 232223 | 242224 | 252225 | ||||
| 131113 | 141114 | 111115 |
A game called tapeworm is played on a board with thirty-three squares. This board can be easily obtained by covering the chessboard with a sheet of cardboard with a cross-shaped cutout.
Useful and exciting entertainment includes composing figures from seven pieces of a square, cut in accordance with Fig. 3, (a), and when composing the given figures, all seven pieces must be used, and they must overlap, even partially, with each other.
In Fig. Figure 4 shows symmetrical figures 1. Try to put these figures together from parts of the square shown in Fig. 3, (a).
(a) (b)
Fig.3
Rice. 4
From the same drawings you can create many other figures (for example, images of various objects, animals, etc.).
A less common version of the game is to make figures from pieces of the square shown in Fig. 3, (b).
Magic squares
Magic square "n 2 -square" let's call a square divided by n 2 cells filled first n 2 natural numbers so that the sums of the numbers in any horizontal or vertical row, as well as on any of the diagonals of the square, are equal to the same numberIf only the sums of numbers in any horizontal and vertical row are the same, then the square is called semi-magical.
Lesson objectives: introduce students to the concept of integer and fractional parts of a number; formulate and prove some properties of the integer part of a number; introduce students to a wide range of uses of the integer and fractional parts of a number; improve the ability to solve equations and systems of equations containing integer and fractional parts of a number.
Equipment: poster “Whoever does and thinks for himself from a young age later becomes more reliable, stronger, smarter” (V. Shukshin).
Projector, magnetic board, algebra reference book.
Lesson plan.
- Organizing time.
- Checking homework.
- Learning new material.
- Solving problems on the topic.
- Lesson summary.
- Homework.
During the classes
I. Organizational moment: lesson topic message; setting the lesson goal; message of the stages of the lesson.
II. Checking homework.
Answer students' questions about homework. Solve problems that caused difficulties when doing homework.
III. Learning new material.
In many algebra problems, you have to consider the largest integer that does not exceed a given number. Such an integer has received a special name “integer part of a number”.
1. Definition.
The integer part of a real number x is the largest integer not exceeding x. The integer part of the number x is denoted by the symbol [x] or E(x) (from the French Entier “antier” ─ “whole”). For example, = 5, [π ] = 3,
From the definition it follows that [x] ≤ x, since the integer part does not exceed x.
On the other hand, because [x] is the largest integer that satisfies the inequality, then [x] +1>x. Thus, [x] is an integer defined by the inequalities [x] ≤ x< [x] +1, а значит 0 ≤ х ─ [x] < 1.
The number α = υ ─ [x] is called the fractional part of the number x and is designated (x). Then we have: 0 ≤ (x)<1 и следовательно, х = [x] + {х}.
2. Some properties of antie.
1. If Z is an integer, then = [x] + Z.
2. For any real numbers x and y: ≥ [x] + [y].
Proof: since x = [x] + (x), 0 ≤ (x)<1 и у = [у] + {у}, 0 ≤ {у}<1, то х+у= [x] + {х} + [у] + {у}= [x] + [у] + α, где α = {х} + {у} и 0 ≤ α <2.
If 0 ≤ α<1. ς о = [x] + [у].
If 1≤ α<2, т.е. α = 1 + α` , где 0 ≤ α` < 1, то х+у = [x] + [у] +1+ α` и
= [x] + [y]+1>[x] + [y].
This property extends to any finite number of terms:
≥ + + + … + .
The ability to find the integer part of a quantity is very important in approximate calculations. In fact, if we know how to find the integer part of the value x, then, taking [x] or [x]+1 as an approximate value of the value x, we will make an error whose value is not greater than one, since
≤ x – [x]< [x] + 1 – [x]=1,
0< [x] + 1– x ≤[x] + 1 – [x] =1.
Moreover, the value of the integer part of the quantity allows you to find its value with an accuracy of 0.5. For this value you can take [x] + 0.5.
The ability to find the whole part of a number allows you to determine this number with any degree of accuracy. Indeed, since
≤ Nx ≤ +1, then
For larger N the error will be small.
IV. Problem solving.
(They are obtained by extracting roots with an accuracy of 0.1 with deficiency and excess). Adding these inequalities, we get
1+0,7+0,5+0,5+0,4 < х < 1+0,8+0,6+0,5+0,5.
Those. 3.1< x <3,4 и, следовательно, [x]=3.
Note that the number 3.25 differs from x by no more than 0.15.
Task 2. Find the smallest natural number m for which
Checking shows that for k = 1 and k = 2 the resulting inequality does not hold for any natural m, and for k = 3 it has a solution m = 1.
This means the required number is 11.
Answer: 11.
Antje in Eqs.
Solving equations with a variable under the “integer part” sign usually comes down to solving inequalities or systems of inequalities.
Task 3. Solve the equation:
Task 4. Solve the equation
By the definition of the integer part, the resulting equation is equivalent to the double inequality
Task 5. Solve the equation 
Solution: if two numbers have the same integer part, then their difference in absolute value is less than 1, and therefore the inequality follows from this equation
And therefore, firstly, x≥ 0, and secondly, in the sum in the middle of the resulting double inequality, all terms, starting from the third, are equal to 0, so x < 7 .
Since x is an integer, all that remains is to check the values from 0 to 6. The solutions to the equation are the numbers 0.4 and 5.
c) marking.
VI. Homework.
Additional task (optional).
Someone measured the length and width of a rectangle. He multiplied the whole part of the length by the whole part of the width and got 48; multiplied the whole part of the length by the fractional part of the width and got 3.2; multiplied the fractional part of the length by the whole part of the width and got 1.5. Determine the area of the rectangle.
Function [ x] is equal to the largest integer greater than x (x– any real number). For example:
Function [ x] has “break points”: for integer values x it “changes abruptly.”
Figure 2 shows a graph of this function, and the left end of each of the horizontal segments belongs to the graph (bold dots), and the right end does not.
Try to prove that if the canonical decomposition of a number n! there is then 
Similar formulas hold for
Knowing this, it is easy to determine, for example, how many zeros the number 100 ends with! Indeed, let it be. Then
And 
.
Therefore, 100! Divided by, i.e. ends with twenty-four zeros.
Figures from square pieces
Useful and exciting entertainment includes composing figures from seven pieces of a square, cut in accordance with Fig. 3, (a), and when composing the given figures, all seven pieces must be used, and they must overlap, even partially, with each other.
In Fig. Figure 4 shows symmetrical figures 1. Try to put these figures together from parts of the square shown in Fig. 3, (a).
From the same drawings you can create many other figures (for example, images of various objects, animals, etc.).
A less common version of the game is to make figures from pieces of the square shown in Fig. 3, (b).
Magic squares
Magic square "n 2 -square" let's call a square divided by n 2 cells filled first n 2 natural numbers so that the sums of the numbers in any horizontal or vertical row, as well as on any of the diagonals of the square, are equal to the same number
If only the sums of numbers in any horizontal and vertical row are the same, then the square is called semi-magical.
The magic 4 2 square is named after Dürer, a 16th-century mathematician and artist who depicted a square in the famous painting “Melancholy.”
By the way, the two lower middle numbers of this square form the number 1514, the date of creation of the painting.
There are only eight nine-cell magic squares. Two of them, which are mirror images of each other, are shown in the figure; the remaining six can be obtained from these squares by rotating them around the center by 90°, 180°, 270°
2. It is not difficult to fully investigate the question of magic squares for n=3
Indeed, S 3 = 15, and there are only eight ways to represent the number 15 as a sum of different numbers (from one to nine):
15=1+5+9=1+6+8=2+4+9=2+5+8=2+6+7=3+4+8=3+5+7=4+5+6
Note that each of the numbers 1, 3, 7, 9 is included in two, and each of the numbers 2, 4, 6, 8 is included in three specified sums, and only the number 5 is included in four sums. On the other hand, out of eight three-cell rows: three horizontal, three vertical and two diagonal, three rows pass through each of the corner cells of the square, four through the central cell, and two rows through each of the remaining cells. Therefore, the number 5 must necessarily be in the central cell, the numbers 2, 4, 6, 8 - in the corner cells, and the numbers 1, 3, 7, 9 - in the remaining cells of the square.