Long ago, when our planet was still young and fiery, three powerful beings known as galactic terraformers transformed it into a beautiful paradise. After completing their work, they left to explore new worlds, leaving behind their source of power: three magical golden hexagons. These hexagons were hidden in dungeons filled with traps and monsters. If someone could gather all three, they would have the power to reshape the world as they wished.
Fast forward thousands of years, and you discover that a wizard named Gordon is trying to collect these hexagons to control the world. Determined to stop him, you embark on a journey through fiery, icy, and sandy lands to find the hexagons first. However, each time you reach a dungeon, you find that a clever merchant named Dongle has beaten you to it. At the end of your quest, you receive an invitation to Dongle’s castle, where you arrive with 99 gems, just like Gordon.
Dongle has not only collected the golden hexagons but has also created five silver hexagons, which are just as powerful. He loves auctions and sets up a bidding war between you and Gordon. The goal is to win a set of three matching hexagons, either all golden or all silver, to gain their power. The auction starts with the golden hexagons, and you and Gordon will take turns bidding on them. If there’s a tie, the winner alternates, starting with you.
You place a bid of 24 gems on the first golden hexagon, and Gordon, using his magic mirror, bids zero, letting you win it easily. Now, you need a strategy to win a set of three hexagons before Gordon does. Should you spend all your gems on the golden hexagons or save some for the silver ones?
Gordon can use his mirror and 99 gems to outbid you on the second golden hexagon. So, how can you make him spend enough on the golds so you can win the silvers? Here’s a hint: if you start the silver auctions with just one more gem than Gordon, you can win. For example, if you have 9 gems and he has 8, you can divide your gems into three groups of three and win all three silver hexagons.
To gain this advantage, imagine Gordon lets you win the second gold auction by tying your bid. You could then bid everything you have left on the third gold hexagon, forcing him to match you to block. If you can bid 51 gems on the third gold, you would enter the silver auctions with a 51 to 48 advantage, ensuring your victory.
If Gordon wins the second gold by bidding 25 against your 24, the total would be 75 to his 74. Neither of you would bid on the third gold since it wouldn’t give either of you three golds. Then, you could bid 25 gems each time in the silver auctions to win three silvers.
In this intense bidding war, your clever strategy and planning allowed you to stay ahead of Gordon. With the silver hexagons in your possession, you now have the power to reshape the world. What will you do with this incredible power?
Imagine you’re on a quest to find hidden hexagons in your school. Create a treasure map with clues leading to different locations where hexagon-shaped tokens are hidden. Work in teams to solve riddles and collect as many tokens as possible. Discuss how teamwork and strategy helped you succeed in this activity.
Participate in a mock auction where you and your classmates bid on items using play money. Practice strategizing your bids to win the most valuable items. Reflect on how your bidding strategy changed throughout the game and what you learned about decision-making.
Using graph paper, design your own dungeon filled with traps and challenges. Include a path to a hidden hexagon. Share your design with classmates and explain the strategies needed to navigate your dungeon successfully. Discuss how you balanced difficulty and fairness in your design.
With a limited number of gems, decide how to allocate them to achieve different goals, such as winning hexagons or saving for future opportunities. Work in pairs to discuss your strategies and compare outcomes. Analyze how different allocations affected your success.
Role-play as characters from the story, such as the wizard Gordon or the merchant Dongle. Engage in a debate about the ethical implications of using the hexagons’ power. Consider different perspectives and articulate your character’s motivations and goals. Reflect on how this activity deepened your understanding of the story’s themes.
According to legend, when this planet was young and molten, three galactic terraformers shaped it into a paradise. When their work was done, they sought out new worlds but left behind the source of their power: three powerful golden hexagons, hidden within dungeons full of traps and monsters. If one person were to bring all three hexagons together, they could reinvent the world however they saw fit.
That was thousands of years ago. Today, you’ve learned of Gordon, a wizard determined to collect the hexagons and impose his will on the world. So you set off on a quest to get them first, adventuring through fire, ice, and sand. Yet each time, you find that someone else got there first—not Gordon, but a merchant named Dongle. At the end of the third dungeon, you find a note inviting you to Dongle’s castle. You arrive with a wallet bursting with the 99 gems you’ve collected in your travels, just moments before Gordon, who also has 99 gems.
Dongle has not only collected the golden hexagons but has also used them to create five silver hexagons, just as powerful as their golden counterparts. Why did Dongle do all this? Because there’s one thing he loves above all else: auctions. You and the wizard will compete to win the hexagons, starting with the three golden ones, making one bid for each item as it comes up. The winners of ties will alternate, starting with you. Whoever first collects a trio of either golden or silver hexagons can use their power to recreate the world.
You’ve already bid 24 gems on the first hexagon when you realize that your rival has a significant advantage: a mirror that lets him see what you’re bidding. He bids zero, and you win the first hexagon outright. What’s your strategy to win a matching trio of hexagons before your rival?
Dongle has presented a challenging dilemma. Do you spend big to try to win the golden hexagons outright? Save as much as possible for silver? Or something in between? Gordon can use his magic mirror and 99 gems to ensure that no matter what you bid on the second gold, he can bid one more and block you. So the real question is—how can you force Gordon to spend enough on the golds to guarantee that you’ll win on the silvers?
Here’s a hint. Let’s say at the start of the silver auctions you had a one-gem advantage, such as 9 to 8. You need to win three auctions, so could you divide your gems into three groups of three and win? For simplicity, let’s assume a set of rules that’s worse for you, where Gordon wins every tie. If you bid 3 each time, the best he could do is win two silver hexagons and have two gems left—which you’ll beat with three bids of 3. Any one-gem advantage where your starting total is divisible by three will lead to victory by the same logic.
So knowing that, how can you force Gordon’s hand in the gold auctions so you go into silver with an advantage? Let’s first imagine that Gordon lets you win the second gold auction by betting some amount X with a tie. You could then bid everything you have left on the third gold hexagon, and he’d have to match you to block. If you could bid 51 on the third gold, you’d go into silver with a 51 to 48 advantage, which you know you can win.
Solving for X reveals that in order to have 51 on round three, you should bid at most 24 on round two. But what about the other possibility, where Gordon wins the second gold against your bid of 24—would this strategy still work? The least he could bid to win the second gold is 25, making the total 75 to Gordon’s 74. No one would then bid on round three since you’ve each blocked the other from getting three golds. After that, you could bid 25 every time to win three silvers.
The bidding war was close, but your ingenuity kept you one step ahead in the chain of inference, and the silver tri-source is yours. Now… what will you do with it?
Hexagons – A six-sided polygon often found in geometric problems and tessellations. – In our math class, we learned how to calculate the area of hexagons by dividing them into triangles.
Gems – Precious or semi-precious stones that can be used as a metaphor for valuable ideas or solutions in problem-solving. – The teacher described the new theorem as one of the hidden gems of geometry.
Bidding – The act of offering a certain amount of resources or effort to achieve a desired outcome, often used in decision-making scenarios. – During the math competition, each team was bidding their time wisely to solve the most problems.
Strategy – A plan of action designed to achieve a specific goal, especially in problem-solving or logical reasoning. – Developing a good strategy is crucial when tackling complex algebraic equations.
Auction – A process of buying and selling items by offering them up for bid, used as an analogy for prioritizing tasks or resources. – In the classroom auction, students had to decide which math problems to solve first based on their difficulty.
Silver – A precious metal often used as a metaphor for second-best solutions or outcomes in problem-solving. – Although we didn’t win the gold, our team was proud to take home the silver in the math olympiad.
Golden – Referring to the best or most valuable solution, often used in optimization problems. – Finding the golden ratio in design can lead to aesthetically pleasing results.
Victory – The achievement of mastering a concept or solving a challenging problem. – Solving the final puzzle in the math challenge felt like a great victory for the team.
Plan – A detailed proposal for achieving a specific goal, especially in solving mathematical problems. – Before starting the geometry project, we created a plan to ensure we covered all necessary topics.
Merchant – A person involved in trade, used as an analogy for exchanging ideas or solutions in collaborative problem-solving. – In our group project, each student acted as a merchant, trading their best ideas to solve the math problem.
