Can You Solve Dongle's Difficult Dilemma?

Imagine a world where three galactic terraformers, after shaping a planet into paradise, left behind the source of their power: three golden hexagons hidden in treacherous dungeons. Legend says that whoever unites these hexagons can reshape the world. But you're not the only one seeking this power.

You find yourself in a race against an evil wizard named Gordon, but a merchant named Dongle has collected the golden hexagons and created five silver hexagons of equal power. Dongle, with a love for auctions, invites you and Gordon to bid for these powerful artifacts. The goal? To acquire a trio of either golden or silver hexagons and wield the power to reinvent the world.

The Auction Begins

You arrive at Dongle's castle, wallet in hand with 99 gems, just as Gordon does. The auction commences, starting with the golden hexagons. You quickly realize Gordon has a significant advantage: a magic mirror that reveals your bids.

The Challenge

Given Gordon's ability to see your bids, how do you strategize to win a matching trio of hexagons before he does?

Cracking the Code

Should you aggressively bid for the golden hexagons, conserve your gems for the silver ones, or find a middle ground? Gordon's magic mirror allows him to outbid you on the second gold hexagon, so the key is to manipulate him into spending enough on the golden hexagons to ensure your victory in the silver auctions.

The Hint

Imagine starting the silver auctions with a slight gem advantage, like 9 to 8. If you need to win three auctions, can you divide your gems into three groups and secure victory? Let's assume a scenario where Gordon wins all ties, making it even more challenging for you.

If you bid 3 gems each time, Gordon can, at best, win two silver hexagons and have two gems remaining. You can then win the final auction with a bid of 3. Any one-gem advantage where your starting total is divisible by three guarantees your win.

The Strategy

How do you manipulate Gordon's bids in the gold auctions to gain that crucial advantage for the silver auctions?

If Gordon allows you to win the second gold auction by tying your bid (let's say X), you could then bid all your remaining gems on the third gold hexagon, forcing him to match your bid to block you. If you could bid 51 on the third gold, you'd enter the silver auctions with a 51 to 48 advantage, ensuring your win.

Solving for X, you realize that to have 51 gems for the third round, you should bid no more than 24 gems on the second round.

The Twist

But what if Gordon wins the second gold against your bid of 24? Would this strategy still work?

The least Gordon could bid to win the second gold is 25, making the totals 75 to your 74. In this case, neither of you would bid on the third gold, as you've both blocked each other from obtaining three golden hexagons. You could then bid 25 gems each time to win three silver hexagons.

Victory

Through careful planning and strategic bidding, you've outmaneuvered Gordon and secured the silver tri-source. The power to reshape the world is now in your hands. What will you do with it?

By anticipating your opponent's moves and calculating the optimal bidding strategy, you can overcome seemingly insurmountable odds and emerge victorious. This puzzle highlights the importance of strategic thinking and adaptability in the face of adversity.