Rolling along the trail
What could be the scores from five throws of this dice?
Age
11 to 14
Challenge level
Problem
A pair of dice is thrown and the score is obtained by finding the product of the two numbers.
In five throws of both dice:
the second score is $5$ more that the first;
the third score is $6$ less than the scond;
the fourth score is $11$ more than the third;
the fifth score is $8$ less than the fourth.
What was the score for each of these five throws?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $10,15,9,20,12$
The possible scores that can be obtained are: $1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36$.
The third and fourth scores differ by $11$. The only pairs of numbers that do this are $(1,12)$, $(4,15)$, $(5,16)$, $(9,20)$ and $(25,36)$.
When these pairs are completed into sequences, they become:
$2,7,1,12,4$
$5,10,4,15,7$
$6,11,5,16,8$
$10,15,9,20,12$
$26,31,25,36,28$
Of these, only the fourth one uses only accessible numbers, so the sequence is $10,15,9,20,12$.