Mr D. R. Kaprekar was an Indian school maths teacher who loved playing with numbers. See if you can follow these steps to find out what he discovered …
Choose a four digit number where the
digits are not all the same (that is not 1111, 2222,…).
Rearrange the digits to get the
largest and smallest numbers these digits can make.
Finally, subtract the smallest number
from the largest to get a new number, and carry on repeating the operation for
each new number.
What do you notice?
Does something similar happen for 3 digit numbers? Can you prove it?
In the 501 game of darts players take turns at throwing 3 darts to reduce their score to zero.
A “checkout” refers to the process of finishing a game by reducing a player’s score to exactly zero, by hitting a double or the bullseye (50 points) with the final dart.
For example, if a player has 40 remaining, they can hit the double 20 (D20) to win.
The maximum checkout is 170. How can you make this?
For which numbers between 140 to 170 can you find a three dart checkout?
140
150
160
141
151
161
142
152
162
143
153
163
144
154
164
145
155
165
146
156
166
147
157
167
148
158
168
149
159
169
Most darts players like to aim to finish on D20, D18, D16 or maybe Bullseye.
A player has 94 left with three darts. They aim for T18
What is their checkout if they hit T18?
What is their next dart if they hit a single 18 instead?
Imagine you have three darts. Would you rather a score of 32 or 30 left? Think about what happens if you just miss D16 and get single 16 left versus just missing D15 and getting single 15.
What checkouts can you find for three darts on 107? What would be the best option do you think?
Here is a pdf with explanations adapted from Better Explained and all the activities below. Solutions to all the activities can be found here.
Decimal
15
20
32
47
50
170
171
141
Hexadecimal
F
14
20
2F
32
AA
AB
8D
Activity 1
Colour in all the hexagons that are multiples of 7 when converted to decimal numbers.
Activity 2
Fill in the table below. You will need to work out what the 3rd column represents in hexadecimal numbers
Decimal
17
88
740
Hexadecimal
11B
FFF
1000
Activity 3
Find the sum of 3A5 + 2D1 by converting to decimal, doing the addition and converting back to hexadecimal.
Can you do it without converting to decimal numbers?
Activity 4
What’s great about binary?
It’s the simplest number system as it only uses two digits – 1 and 0. This means it is very easy to build in hardware. You just need things that can turn on or off (representing 1 and 0), so it is fundamental to all computers. For example, in a transistor, ‘0’ means no electricity is flowing, whilst ‘1’ means there is a flow of electricity.
How are binary and hexadecimals linked?
Because one block of four binary numbers represents one hexadecimal number it is easy for programmers to visualise numbers in hexadecimal. Programmers sometimes use words written in hexadecimal in their programs.
Take the sequence above. Cross off all the numbers in the middle that share a factor with either 24 or 35. You should be stuck with 29 and 31 not crossed out.
Can you choose a sequence where you can cross out all the numbers in the middle? It can start anywhere and be any length.
