Seven Dwarfs
The Seven Dwarfs have seven different birthdays. How old can they be?
Age
11 to 14
Challenge level
The Seven Dwarfs were born on the same day, in seven consecutive years. The ages of the youngest three add up to $42$ years. What do the ages of the oldest three add up to?
Answer: 54
Symbols
Suppose the youngest dwarf is $x$ years old.
Sum of the ages of the youngest three is $x+(x+1)+(x+2)=3x+3$
Sum of the ages of the oldest three is $(x+4)+(x+5)+(x+6)=3x+15$.
We know that $3x+3=42$, so $3x+15=42+12=54$, so the ages of the oldest three of the Seven Dwarfs add up to $54$.
Patterns in the numbers
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