Knapsack

You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value.

Age

14 to 16

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

This problem is based on the idea of Knapsack codes.

You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value that is given in the lookup table below.

You go off on a camping trip with 5 sticks in your knapsack. They have lengths of 1,3,5,10 and 20 centimetres, these will help you decode any message your friend sends.

A coded message arrives. It is the number 31. To decode the message you must work out which sticks you need to make a length of 31cm and convert this information into the binary code that tells you the letter. Decoding will be easy because your sticks form part of a superincreasing series (each stick is longer than the sum of the lengths of all the smaller sticks).

Taking the largest length off first leaves 31-20 = 11 so the coded letter used the 20 cm stick. 11-10 = 1 so the coded letter also involves the 10cm stick. With 1 left the 5-stick and the 3-stick are not used, just the 1-stick.

This gives you a binary code of 10011 (1x1cm+0x3cm+0x5cm+1x10cm+1x20cm).

So using the binary lookup table, the number 31 represents is 10011, which is the letter s.

Lookup Table

Letter	Binary Reference	Letter	Binary Reference
a	00001	n	01110
b	00010	o	01111
c	00011	p	10000
d	00100	q	10001
e	00101	r	10010
f	00110	s	10011
g	00111	t	10100
h	01000	u	10101
i	01001	v	10110
j	01010	w	10111
k	01011	x	11000
l	01100	y	11001
m	01101	z	11010

Using the sticks in your knapsack decode the message: 33,18, 20, 1, 31, 20, 30, 33.

That was easy, but say your knapsack code involved a non-superincreasing series that used 1,2,3,4,5cm sticks.

Using this knapsack can you decode the message 1, 5, 14, 4, 5, 8, 10, 5, 4, 7, 9?

Can you explain why superincreasing series are so much easier to decode?

Student Solutions

We had a number of solutions to this problem and I would like to mention Karan of Beecroft Primary school and Ross of the Blue Coat School, who both managed to get the first part right but did not quite get to grips with the second.

Below is a very good solution sent by Andrei of School 205 Bucharest. Well done Andrei.

As in the first case I have to deal with a super increasing series, for each coded number there is only one corresponding letter:

Code	Binary	Letter
33	01011	K
18	01110	N
20	00001	A
1	10000	P
31	10011	S
20	00001	A
30	00011	C
33	01011	K

So, the coded word is "knapsack". For, the second part of the problem, I take each number and write it in all the possible combinations, which gives us several letters to chose from:

Code	Sum	Binary	Letter
1	1	10000	p
5	2+3	01100	l
	4+1	10010	r
	5	00001	a
14	2+3+4+5	01111	o
4	1+3	10100	t
	4	00010	b
5	2+3	01100	l
	4+1	10010	r
	5	00001	a
8	1+2+5	11001	y
	1+3+4	10110	v
	3+5	00101	e
10	1+4+5	10011	s
	1+2+3+4	11110	-
	2+3+5	01101	m
5	2+3	01100	l
	4+1	10010	r
	5	00001	a
4	1+3	10100	t
	4	00010	b
7	1+2+4	11-1-	z
	3+4	00110	f
	2+5	01001	i
9	1+3+5	10101	u
	4+5	00011	c

There are many possibilities to make a word. But I need to find one which has sense. The possibilities are written in the following table:

p	l	o	t	l	y	s	l	t	z	u
	r		b	r	v	m	r	b	f	c
	a			a	e		a		i

The only solution that I can find is: "problematic".

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners

Knapsack

Problem

Getting Started

Student Solutions

Teachers' Resources

Notes: