After months of travel, you finally arrive at Duonia, the renowned city that houses the temple, your pilgrimage’s ultimate destination. Entering from the northwest, you pass through the city gates and the welcome center, where you receive a map and a brochure. The map reveals that Duonia is laid out in a grid of 16 blocks, formed by five streets running west to east and five more intersecting them from north to south. Standing on the northernmost street facing east, you find the blocks containing the gate and the welcome center behind you. The temple’s sole entrance is located at the very southeast corner of the city.
Reaching the temple isn’t just a matter of walking; there’s a unique tax imposed on all visitors. This tax, which must be paid upon arrival at your destination, starts at zero and increases by two silver for every block you walk east. It doubles for every block you walk south. However, a recent reform aimed at fairness halves your total bill for every block you walk north and subtracts two silver for every block you walk west. Just by passing through the gate and welcome center, you already owe four silver. As a pilgrim without money and unable to earn any, you face a conundrum. Moreover, the rules of your pilgrimage forbid you from walking over any stretch of ground more than once, although you can cross your own path. Can you reach the temple without owing any tax or retracing your steps?
Examining the map, you realize that moving towards the temple always increases the tax, while moving away decreases it, making it seem impossible to reach the temple without incurring a debt. However, what if you walk around a single block? Starting with a debt of four silver and moving clockwise, your tax bill becomes six, then 12, then 10, and finally five. Looping again, you would owe seven, 14, 12, and six. Each clockwise loop leaves you with an extra silver owed. Conversely, a counterclockwise loop starting with four silver changes your bill to eight, 10, five, and three. Looping again, you owe six, eight, four, and two. Each counterclockwise loop actually reduces your debt by one silver. This is because the order of tax application makes a significant difference, even if the taxes for opposite directions seem to balance each other out.
Starting with a debt of four silver, four counterclockwise loops would theoretically bring you down to zero. However, the restriction against walking the same block twice complicates this approach. Instead, you can reduce your bill by taking one large counterclockwise loop through the city. From your starting position, walk three blocks south, leaving the southernmost street clear for the final stretch. Continue counterclockwise by going east for two blocks, which increases your debt to a staggering 36 silver. Now, you can begin reducing your bill. Walking three blocks north and one block west cuts your debt to 2.5 silver. Unable to go west without trapping yourself, you head one block south and then three blocks west, resulting in a debt of -1 silver. Since doubling a negative number still yields a negative, walking the three blocks to the south wall means the city owes you eight silver. Fortunately, this is just enough to cover the final blocks to the temple.
As you enter the temple, you reflect on the lesson learned from your pilgrimage: sometimes, an indirect route is the best way to reach your destination. This journey through Duonia not only tested your problem-solving skills but also reinforced the value of strategic thinking and adaptability.
Using graph paper, draw a 5×5 grid to represent the city of Duonia. Mark the starting point at the northwest corner and the temple at the southeast corner. Plan a route that minimizes your tax bill while adhering to the rules. Share your route with classmates and discuss different strategies.
Create a game where you and your classmates take turns moving a token on a grid. Each move should follow the tax rules described in the article. Keep track of your tax bill on a scorecard. The goal is to reach the temple with the lowest possible tax. Reflect on the strategies that worked best.
Form small groups and role-play the journey to the temple. Assign roles such as the pilgrim, tax collector, and city guide. Use props like maps and tokens to navigate the city. Discuss the challenges faced and how teamwork and strategic thinking helped overcome them.
Write a series of math story problems based on the tax rules and movements described in the article. Exchange problems with a partner and solve them. This activity will help reinforce your understanding of the tax calculations and the impact of different movements.
Write a short story from the perspective of the pilgrim. Describe the journey through Duonia, the challenges faced, and the strategies used to reach the temple. Focus on the lessons learned about problem-solving and adaptability. Share your story with the class.
Tax – A percentage of income or the value of goods that is paid to the government. – The government uses the tax collected from citizens to fund public services like schools and roads.
Silver – A shiny, grayish-white metal that is often used in jewelry and coins. – In math class, we learned that the value of silver can change based on supply and demand.
Blocks – Solid pieces of material, often used in building or as a unit of measurement. – We used blocks to create a model of a city and calculate the area of each building.
Map – A visual representation of an area that shows the relationship between different locations. – To find the quickest route to the park, we used a map to plan our journey.
Debt – Money that is owed to someone else. – Understanding how to manage debt is important for making smart financial decisions.
Clockwise – In the direction that the hands of a clock move. – To rotate the shape clockwise, we turned it to the right.
Counterclockwise – In the opposite direction of the movement of the hands of a clock. – When we turned the dial counterclockwise, we were able to unlock the combination.
Journey – A trip or travel from one place to another. – Our journey to learn new math concepts is filled with challenges and discoveries.
Problem-solving – The process of finding solutions to difficult or complex issues. – In math class, we practiced problem-solving by working through challenging equations together.
Strategy – A plan of action designed to achieve a specific goal. – Developing a strategy for solving math problems can help you work more efficiently.
