# Problem of the Week Problem D and Solution Fast Bikers

## Problem

The top five finishers in a bike race were Albine, Farrah, Jasna, Nuan, and Terese, in some order. They all rode a different-coloured bike (black, white, green, blue, or red), and were a different age $$(18,~21,~25,~26,$$ or $$29)$$. Use the clues below to determine each personâ€™s age, bike colour, and final position in the race.

1. The five people in the race were the person who finished first, the $$26$$-year old, Nuan, the person with the white bike, and Albine.

2. Jasna rode the white bike and finished third.

3. Nuan is $$3$$ years younger than the person who rode the red bike.

4. The oldest person rode the white bike and finished just before the youngest person, who rode the black bike.

5. The person with the green bike finished first, and someone younger finished right after.

6. Terese finished right after Nuan.

You may find the following table useful in organizing your solution.

## Solution

First we will give the answer, and then an explanation as to how we arrived at the answer.

• Farrah is $$25$$ years old, rode the green bike, and finished first.

• Albine is $$21$$ years old, rode the red bike, and finished second.

• Jasna is $$29$$ years old, rode the white bike, and finished third.

• Nuan is $$18$$ years old, rode the black bike, and finished fourth.

• Terese is $$26$$ years old, rode the blue bike, and finished fifth.

There are many different ways to arrive at the answer above. You may have used the table provided to keep track of matches that were confirmed or deemed impossible while examining and combining the different clues. Below we present an explanation in words only. It may be helpful to follow along by filling in the table as you read the explanation.

From clues $$2$$ and $$4$$, Jasna rode the white bike, finished third, and is $$29$$ years old. Clue $$4$$ also tells us that the $$18$$-year old rode the black bike and finished fourth.

From clue $$3$$, Nuan and the person who rode the red bike must be either $$18$$ and $$21$$ years old, or $$26$$ and $$29$$ years old. However, from clue $$1$$, we know Nuan is not $$26$$ years old. It follows that Nuan must be $$18$$ years old and the person who rode the red bike must be $$21$$ years old. That means Nuan is the person who finished fourth and rode the black bike.

From clue $$5$$, Terese finished fifth. It follows from clue $$1$$ that Terese is $$26$$ years old since she did not finish first or ride the white bike.

The people who finished first and second must therefore be $$25$$ and $$21$$ years old, in some order. It follows from clue $$5$$ that the $$25$$-year old must have finished first, followed by the $$21$$-year old, who we already determined rode the red bike. Clue $$5$$ also tells us the person who finished first rode a green bike. It follows that Terese rode a blue bike.

From clue $$1$$, Albine did not finish first, so that means she finished second, and Farrah finished first.