Enrichment in Mathematics – Algebra
Solving Algebraic Puzzles
Teachers Notes
This problem brings in the use of equations to find two unknown quantities.
It could be used as a starter or plenary, or as an extension/enrichment problem
solving activity.
Pupils should be given the initial information and asked for a solution.
It would then be useful to discuss various methods that pupils may have used to get
there. They may initially use trial and improvement, but more sophisticated methods
involving equations & substitution can be shown to be quicker and more efficient. The
main aim is for the pupils to discover the best way of solving the problem themselves,
rather than being shown a method.
Solutions
Puzzle A
using equations:
from 1:
=7–
subst. in 2: (7 – ) ×  = 12
² – 7 + 12 = 0
( – 3) ( – 4) = 0
Puzzle B
using equations:
 +  = 21
 ×  = 80
from 1:
 = 21 – 
subst. in 2: (21 – ) ×  = 80
² – 21 + 80 = 0
( – 5) ( – 16) = 0
Puzzle C
using equations:
 +  = 120
7 = 
subst. in 1:  + 7 = 120
8 = 120
Puzzle D
++=6
××=6
answers: 3, 4 (either way round)
method: number-facts recall
(or trial & imp)
answers: 5, 16 (either way round)
method: trial & improvement
(or substitution)
answer: 105 correct (15 incorrect)
method: trial & improvement
(or substitution)
answers 1, 2, 3
method: trial & improvement
NB this time there are 2 equations/facts and 3 unknowns so forming equations helps
to represent the problem but does not otherwise help to solve it.
This is an example of a happy number: the product of the factors (other than the
number itself) is equal to their sum
Worcestershire Numeracy Team
Enrichment Activities
Enrichment in Mathematics – Algebra
Solving Algebraic Puzzles
Puzzle A
+=7
 ×  = 12
Puzzle B
Find the two numbers.
I am thinking of two numbers.
The product of my numbers is 80.
The sum of my numbers is 21.
Find the two numbers.
Puzzle C
On an algebra test, I had seven times as many correct
answers as incorrect ones.
There were 120 questions on the test.
How many did I get right?
Puzzle D
The sum and the product of three numbers is 6.
Find the three numbers.
In pairs or small groups, compare your methods for
solving each of the puzzles.
Decide between you on ONE best method for each puzzle,
explaining why it is the best.
Worcestershire Numeracy Team
Enrichment Activities