Teaching Schoolchildren to Solve Problems on the Topic
"Number Systems"
MICHAEL DOLINSKY
Faculty of Mathematics and Programming Technologies,
Francisk Skorina Gomel State University,
Sovetskaya Str.104, 246019, Gomel,
BELARUS
Abstract: - The author has been teaching programming to Gomel schoolchildren for more than four decades. For
the last twenty-five years, training has been focused primarily on preparing for programming competitions from
school to international competitions (IOI). Since 1997, 15 students of the author have won a total of 26 medals at
IOI 1997 - 2023. The article contains a description of the teaching methods and tools used by the author to teach
the topic “Number Systems”. An essential technical basis for training is the instrumental distance learning system
DL.GSU.BY, was created and developed under the leadership of the author from 1999 to the present time. In this
article, the following questions are sequentially considered (with the solution of the corresponding original
problems from the American and Russian Internet Olympiads): binary number system, binary counting, converting
numbers from decimal to binary, and vice versa. Hexadecimal and octal number systems. Conversion of numbers
from one number system to another.
Key-Words: - computer science olympiads, instrumental system of distance learning, programming, DL.GSU.BY,
secondary school, high school, number systems, problem solving.
Received: March 19, 2024. Revised: October 11, 2024. Accepted: November 8, 2024. Published: December 10, 2024.
1 Introduction
Increasing the effectiveness of programming classes
largely depends on the availability of literature
corresponding to the level of training of students.
Therefore, they are actively published books, [1], [2],
[3], [4], [5] with system explanations of set of
algorithms as well as articles devoted to concrete
themes such as loops [6], numeral systems [7],
Chinese remainder theorem [8], sorting [9], recursion
[10], [11], dynamic programming [12], [13], graphs
[14], [15], maximum flow [16], binary indexed tree
[17], wavelet tree [18], string hashing [19].
2 Problem Formulation
It has been very accurately noted that the study of
programming automatically leads to an increase in
motivation for doing mathematics, as well as to an
improvement in the quality of assimilation of
mathematical knowledge, [20]. This is especially true
when the topics of study in programming are closely
related to mathematical concepts. One of these topics
is the topic "Number systems". In parallel with
studying the theoretical foundations of the topic, it is
also important to develop computational thinking
[21] and assess its quality [22]. The study of
programming in Gomel begins for those interested in
the first grade, so in grades 5-8 there are children
who have approached the topic “Number Systems”
according to the curriculum, but, of course, books
intended for university and high school students are
not suitable for their learning.
3 Problem Solution
The author has been teaching programming to
schoolchildren of different ages for many years [23]
based on the distance learning instrumental system
DL.GSU.BY [24], developed under the supervision
of the author. All this time, the author has been
creating literature for self-study by schoolchildren,
trying to present the material in the simplest, clearest,
and most understandable form possible. This article
provides an example of such material for learning to
solve problems in computer science on the topic of
"Number systems". Such material may be of interest
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to teachers both as an illustration of the teaching
methodology and in terms of content. At the same
time, the author believes that this material can be
very useful and interesting for both schoolchildren
and students involved in self-study. All those
interested are invited to the following order of work:
put the article aside and try to independently
complete the proposed task after reading the
conditions of the task.
3.1 Binary Number System
People are accustomed to counting in the decimal
number system, making up all numbers from the
digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. For example, 349 is 3
hundreds, 4 tens, 9 units. It is known from history
that this is due to the presence of 10 fingers on the
hands of a person. In this sense, it turned out that
“computers have only two fingers” and, accordingly,
only two numbers: 0 and 1. And it is from these
numbers that all numbers are made up. Arithmetic in
binary system is simplified. Consider addition:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (0 and 1 carry to the next bit)
Now that, knowing the rules of addition, let's try to
count in the binary system.
The original number is 0.
0 + 1 = 1
1 + 1 = 10 (that's 2 in decimal, by the way)
10 + 1 = 11 (3 in decimal)
10
+
1
---
11
11+1 = 100 (4 decimal)
1 (carry from LSB)
11
+ 1
-------
100
100 + 1 = 101 (5 decimal)
100
+
1
------
101
Fill in the binary counting table yourself from 0
to 15 and check with Table 1.
Table 1. Decimal and binary counting from 0 to 15
0
0
1
1
2
10
3
11
4
100
5
101
6
110
7
111
8
1000
9
1001
10
1010
11
1011
12
1100
13
1101
14
1110
15
1111
How to check the correct representation of a
number in the binary system?
In the decimal system, the weights of digits are
the powers of the number 10: from right to left (from
the least significant digit to the most significant) 1,
10, 100 (10 * 10), 1000 (10 * 10 * 10), etc. In the
binary system, the weights of digits are the powers of
the number 2: 1, 2, 4 (2 * 2), 8 (2 * 2 * 2) ... (from
right to left, from the least significant digit to the
most significant).
For example, 1101 = 1 * 8 + 1* 4 + 0* 2 + 1 * 1 =
13.
Okay, we have learned to count in the binary
system. And now we want to know the binary
representation of the number 130 (and not 13, which
is already in the table). Is it really necessary to count
in binary to 130? No, not necessarily, you can use the
algorithm described below.
In the general case, to convert the number X
from decimal to binary, it is enough to sequentially
divide by 2 the number X and the resulting quotients,
and write the remainder as digits of the binary
number from low to high. For example, for the
number 130 (Table 2).
Table 2. An example of converting the number 130
from decimal to binary
Quotient
Remainder
Current
Response
130
130/2
65
0
0
65
65/2
32
1
10
32
32/2
16
0
010
16
16/2
8
0
0010
8
8/2
4
0
00010
4
4/2
2
0
000010
2
2/2
1
0
0000010
1
1/2
0
1
10000010
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Below is a code fragment in the Pascal
programming language that converts the number x
into binary representation (into the string variable s).
var
x : longint;
s : string
begin
readln(x);
if x=0 then s:='0' else s:='';
while x<>0 do
begin
if odd(x)
then s:='1'+s
else s:='0'+s;
x:=x div 2; {x:= x shr 1;}
end;
writeln(s);
end.
Here Odd(x) is a standard Pascal function that
returns "true" (true) if the number x is odd (that is,
the remainder of its division by 2 is 1) and the value
"false" (false) otherwise (that is, the remainder of
dividing the number x by 2 is 0).
In general, the division is a rather time-
consuming arithmetic operation. However, division
by 2 can be done much faster by using a right shift
operation by 1 bit:
x:=x shr 1;
Let's, for example, see what happens to the
decimal number 13 (in binary form 1101) if we shift
it to the right by 1 digit:
1101 -> 0110
(the bit being pushed out is lost, a 0 is pushed in).
0110 is 6 (Table 2). That is, when shifting to the
right, the number 13 was indeed divided by 2 (the
quotient is 6, the remainder is 1).
3.2 BINNUM Problem (USA Computing
Olympiad 2005 January Bronze)
Given a positive integer A (0 <= A <=
2,110,000,000), print its binary (base two)
representation (without extra leading zeroes, of
course). Do not use automatic conversion routines or
automatic binary printing routines.
INPUT FORMAT:
* Line 1: An integer
SAMPLE INPUT (file binnum.in):
277309
OUTPUT FORMAT:
* Line 1: The binary representation of the input
integer
SAMPLE OUTPUT (file binnum.out):
1000011101100111101
Complete solution in Pascal programming language:
var
x : longint;
s : string
begin
assign(input,'binnum.in');
reset(input);
assign(output,'binnum.out');
rewrite(output);
readln(x);
if x=0 then s:='0' else s:='';
while x<>0 do
begin
if odd(x)
then s:='1'+s
else s:='0'+s;
x:= x shr 1;
end;
writeln(s);
close(input); close(output);
end.
3.3 Countable Numbers Problem (USA
Computing Olympiad 2005 Open
Bronze)
Cow hooves have no fingers to speak of, so they
count in binary instead of decimal. Of course, with
only four hooves, they can't count so very high in
normal binary.
They've devised a clever counting method,
though. They stand on one to four of their legs after
putting marks in the dirt. The marks represent binary
0's; their leg locations represent binary 1's. This
means that they can represent numbers whose binary
representation has four or fewer 1's.
Given a range of positive integers (both the start
number and stop number are in the range
1..15,000,000), determine how many numbers in that
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range the cows can represent with four or fewer one
bits.
INPUT FORMAT:
* Line 1: Two space-separated numbers: the
beginning and end of a range to be checked
SAMPLE INPUT (file cnums.in):
100 105
OUTPUT FORMAT:
* Line 1: The count of numbers in the range that can
be represented by four or fewer 1's using binary
arithmetic.
SAMPLE OUTPUT (file cnums.out):
five
OUTPUT DETAILS:
Num Binary Number of 1's Representable by
cows?
100 1100100 3 Yes
101 1100101 4 Yes
102 1100110 4 Yes
103 1100111 5 No
104 1101000 3 Yes
105 1101001 4 Yes
Recommendations for solving the problem.
In the specified range, you need to check how many
units the binary representation of the number
contains. If the units is less than or equal to 4, the
number fits, otherwise it doesn't. To count units in
the binary representation of a number, it is
convenient to use the above-described standard
algorithm for converting a number to a binary
system, in which the accumulation of binary digits is
replaced by counting ones. Below is the full text of
the program that solves this problem.
var
k,b,e,i : longint;
function Good(x:longint):boolean;
var
z : longint;
begin
z:=0;
while (x<>0) and (z<=4) do
begin
if odd(x) then inc(z);
x:=x shr 1;
end;
Good:= z<=4;
end;
begin
assign(input,'cnums.in');
reset(input);
assign(output,'cnums.out');
rewrite(output);
readln(b, e);
k:=0;
for i:=b to e do
if Good(i) then inc(k);
writeln(k);
close(input); close(output);
end.
3.4 Hexadecimal and Octal Number Systems
In the Pascal programming language, 32 binary digits
are allocated to store the value of a variable of type
Longint (in other words, they also say this: 32 bits, or
32 binary digits). It is very difficult for a human,
unlike a computer, to read and understand 32-bit
sequences of binary numbers. Therefore, to reduce
the representation of such numbers, the hexadecimal
and 8-decimal number systems were invented. There
are 16 digits in the hexadecimal number system: 0..9,
A,B,C,D,E,F (Table 3).
Table 3. Decimal, binary and hexadecimal counting
from 0 to 15
10
2
16
0
0000
0
1
0001
1
2
0010
2
3
0011
3
4
0100
4
5
0101
5
6
0110
6
7
0111
7
8
1000
8
9
1001
9
10
1010
A
11
1011
B
12
1100
C
13
1101
D
14
1110
E
15
1111
F
To convert a number from the binary system to
hexadecimal, the sequence of binary digits is divided
from RIGHT into tetrads (groups of binary digits of 4
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digits), and each tetrad is replaced by the
corresponding hexadecimal digit (Table 3).
For example
11110000101001010111111000000000 =
1111 0000 1010 0101 0111 1110 0000 0000 =
F0A57E00
To convert a number from the hexadecimal
system to binary, according to the same table,
hexadecimal digits are replaced with binary tetrads,
for example:
BA34EF1D = 1011 1010 0011 0100 1110 1111 0001
1101 =
10111010001101001110111100011101
The octal system has 7 digits from 0 to 7 (Table 4).
Table 4. Decimal, binary, and octal counting from 0
to 7
10
2
8
0
000
0
1
001
1
2
010
2
3
011
3
4
100
4
5
101
5
6
110
6
7
111
7
To convert a number from the binary system to
octal, the sequence of binary digits is divided RIGHT
into triads (groups of binary digits of 3 digits), and
each triad is replaced by the corresponding octal digit
(Table 4). For example:
11 110 000 101 001 010 111 111 000 000 000 = 3 6
0 5 1 2 7 7 0 0 0 = 36051277000
To convert a number from the 8-ary system to
binary, according to the same table, 8-digit digits are
replaced by binary triads, for example:
32073702145 = 3 2 0 7 3 7 0 2 1 4 5 =
011 010 000 111 011 111 000 010 001 100 101 =
11010000111011111000010001100101
(the leading non-significant zero is discarded).
3.5 Hexadecimal Conversion Problem. (USA
Computing Olympiad 2010 March
Bronze)
Bessie was teaching Jessie, her protege interested in
programming contests the binary facts of life. She
explained that computers work in binary (base 2) and
that all computer numbers in general are stored as 0's
and 1's. Jessie was a bit unclear on the concept, so
Bessie made her an exercise, shown below.
Write a program that converts an unsigned
hexadecimal number to octal (base 8) form.
Hexadecimal number can have at most 100,000 digits
and is composed of digits and capital letters from A
to F.
Note: a hexadecimal number is a special way of
representing numbers in base 16. The digits 0-9 still
correspond to 0-9, and then A (capital A!)
corresponds to 10, B to 11, etc. (F stands for 15).
For example, the hexadecimal number A10B
corresponds to the (decimal) number
11*1+0*16+1*16^2+10*16^3 = 41227. The
corresponding octal (base 8) number would be
120413, since 3
*1+1*8+4*8^2+0*8^3+2*8^4+1*8^5 = 41227.
Hint: there is an easier way to convert from
hexadecimal to octal than by converting hexadecimal
-> decimal -> octal. It might help to think about the
numbers in binary (base 2).
INPUT FORMAT:
* Line 1: A single hexadecimal number. Multidigit
numbers will have no leading zeroes (ie, A1 instead
of 00A1). 0 (by itself) is a valid input.
SAMPLE INPUT (file hex.in):
123ABC
OUTPUT FORMAT:
* Line 1: The octal value with no leading zeros. If the
input is 0, the output should also be 0.
SAMPLE OUTPUT (file hex.out):
4435274
Recommendations for solving the problem:
1.It is enough to convert the input sequence of
hexadecimal digits to binary, and then to octal.
2.To work with such long strings (100,000
characters), you can use the ansistring type (Free
Pascal). The following is the full text of the Free
Pascal solution to the problem.
const
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M162 : array [1..16] of string = (
'0000', '0001', '0010', '0011',
'0100', '0101', '0110', '0111',
'1000', '1001', '1010', '1011',
'1100', '1101', '1110', '1111' );
d16 = '0123456789ABCDEF';
var
s16,s8,s2 : ansistring;
c : string
i,d,k : longint;
function c8 (c :string) : string;
begin
if c='000' then c8:='0';
if c='001' then c8:='1';
if c='010' then c8:='2';
if c='011' then c8:='3';
if c='100' then c8:='4';
if c='101' then c8:='5';
if c='110' then c8:='6';
if c='111' then c8:='7';
end;
begin
assign(input,'hex.in'); reset(input);
assign(output,'hex.out');
rewrite(output);
readln(s16);
d:=length(s16);
s2:='';
for i:=1 to d do
s2:=s2+m162[pos(s16[i],d16)];
k:=4*d;
while k>=3 do
begin
c:=s2[k-2]+s2[k-1]+s2[k];
s8:=c8(c)+s8;
dec(k,3);
end;
if k=1 then s8:=c8('00'+copy(s2,1,k))+s8;
if k=2 then s8:=c8('0' +copy(s2,1,k))+s8;
if (s8[1]='0') and (s8<>'0') then delete(s8,1,1);
writeln(s8);
close(input); close(output);
end.
From the previous material, it is known how to
convert a number from the decimal system to
hexadecimal or octal, through the binary system. But
there is a way to convert the number n from the
decimal number system to any system with base k, in
the same way as it was done when converting to the
binary system - by dividing by the base of the
number system and storing the remainders in reverse
order:
The following is a fragment of such a program
for k from 2 to 9
readln(n, k);
s:=''; p:=1;
while n<>0 do
begin
x:=n mod k;
s:=chr(x+ord('0'));
n:=n div k;
end;
writeln(s);
For number systems with bases greater than 10,
numbers starting from 10 are denoted by Latin letters
A, B, C, ... And the fragment of the corresponding
program changes somewhat:
readln(n, k);
s:=''; p:=1;
while n<>0 do
begin
x:=n mod k;
if x<10
then s:=chr(x+ord('0'))
else s:=chr(x-10+ord('A'));
n:=n div k;
end;
writeln(s);
Note: when checking the correctness of writing
programs for converting numbers from one number
system to another, which are powers of two: binary,
octal, hexadecimal, it is convenient to use the
Windows calculator (to start, just type calc in the
program launch line).
3.6 Problem Easy Calculation (2nd Russian
Internet Olympiad 2006)
Given a natural number n. It is necessary to translate
it into a k-ary number system and find the difference
between the product and the sum of its digits in this
number system. For example, let n = 239, k = 8.
Then the octal representation of n is 357, and the
answer to the problem is 3 * 5 * 7 - (3 + 5 + 7) = 90.
Input format: The input file contains two natural
numbers: n and k (1 <= n <= 10^9; 2 <= k <= 10).
Both of these numbers are given in the decimal
number system.
Output format: Output the answer to the problem (in
decimal notation) into the output file.
Input example: Output example:
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239 8 90
Input example: Output example:
1000000000 7 -34
Below is the text of the program that solves this
problem.
var
s,p,n,k,x : longint;
begin
readln(n, k);
s:=0; p:=1;
while n<>0 do
begin
x:=n mod k;
inc(s, x);
p:=p*x;
n:=n div k;
end;
writeln(ps);
end.
3.7 Training Methodology
The described problems are part of a package of
problems on the topic “Number Systems”. The
student is asked to solve problems in order of
increasing difficulty. The DL system provides
automatic verification of solutions in the
programming languages Pascal, C++, Python, Java,
C#, and the original author’s set of tests. If a
student’s solution on some test produces an answer
that is not equal to the author’s, the student can take
the input data and the author’s answer on this test
from the DL system, find an error in his program,
correct it and send it for testing again. If a student
cannot develop a complete solution to a problem, he
receives a teacher's description of the solution to the
problem, including the source text of the correct
solution in Pascal, parses it, and, if necessary, can
consult with other students or the teacher. Then he
marks what is needed in his notebook. And he always
rewrites the solution without using the notes. More
precisely, he can use notes, but in this case, he must
rewrite the solution from a blank screen, and so on
until he writes the solution completely himself, from
the first to the last letter.
4 Conclusion
This article provides material for learning to solve
problems in computer science on the topic of
"Number systems". In particular, the decimal, binary,
hexadecimal, and octal number systems and ways of
translating numbers between them are considered.
The technical basis of the methodology is the
developed instrumental system of distance learning
(Distance Learning Belarus - http://dl.gsu.by). All
tasks given in the article can be passed in the course
"Algorithmization Methods".
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DOI: 10.37394/232010.2024.21.20
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Contribution of Individual Authors to the
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WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2024.21.20
Michael Dolinsky
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