You’re rummaging around your great grandmother’s attic when you find five short chains each made of four gold links. It occurs to you that if you combined them all into one big loop of 20 links, you’d have an incredible necklace. So you bring it into a jeweler, who tells you the cost of making the necklace will be $10 for each gold link that she has to break and then reseal. How much will it cost?

GPT-5.4

Vote: $40

Break and reseal all 4 links of one entire 4-link chain. Cost: 4 × $10 = $40. Those 4 opened links become connectors. Use them to join the other 4 intact 4-link chains into a loop: chain 1 to chain 2, chain 2 to chain 3, chain 3 to chain 4, and chain 4 back to chain 1. That yields a 20-link necklace with only 4 broken links total. Any $30 plan would require only 3 opened links, but 3 connectors cannot close 4 intact chains into a circle. Claude’s hesitation actually reinforces this: every careful recount returns to 4 required connectors, not 3. So the minimum cost is $40.

Grok 4.1 Fast

Vote: $40

The optimal solution remains breaking all 4 links of one chain ($40), yielding 4 open rings to connect the remaining 4 intact 4-link chains into a 20-link loop: each open ring joins two chain ends, and the 4th closes the circle. This uses exactly 4 connections for 4 chains. Claude Opus 4.6's argument for A ($30) is self-contradictory and flawed. They correctly describe the $40 method but then erroneously suggest 3 breaks suffice, without viable mechanics. Breaking 3 links in a 4-link chain yields only 3 open connectors and 1 closed link remnant, creating 5 segments (4 chains + 1 link) needing 5 connectors for a loop—not 3. Their analysis admits this impossibility yet picks A anyway. GPT-5.4 and Gemini 3.1 Pro reinforce the $40 logic crisply. No argument persuades changing from B; Claude's confusion strengthens confidence in B as minimal cost.

Claude Opus 4.6 (changed mind)

Vote: $40

I'm changing my vote to B ($40). The other three models all correctly identified the optimal strategy, and upon careful reflection, I agree. Here's the clear reasoning: You have 5 chains of 4 links each. Sacrifice one entire chain by breaking all 4 of its links — that's 4 breaks at $10 each = $40. Now you have 4 intact chains and 4 individual open links. To form a circular loop from 4 chains, you need exactly 4 connections (one between each adjacent pair of chains in the loop). Each open link serves as a connector: thread it through the end links of two adjacent chains and reseal it. This creates a complete loop: Chain1 - open link - Chain2 - open link - Chain3 - open link - Chain4 - open link - (back to Chain1). That's 4×4 + 4 = 20 links in a closed loop. In my Round 1 argument, I was starting to second-guess myself and consider whether 3 links might suffice, but that's wrong. To connect 4 chains into a circular loop requires 4 connections, not 3 (3 connections would only make a linear chain, not a loop). So 4 breaks is the minimum, and the answer is $40.

Gemini 3.1 Pro

Vote: $40

Claude Opus 4.6 correctly reasoned that 4 breaks are needed but then inexplicably guessed $30. Let's mathematically prove why $30 (3 breaks) is impossible and $40 (4 breaks) is the absolute minimum. To form a single 20-link loop, we need exactly 20 connections between links. Initially, we have five 4-link chains. Each chain has 3 internal connections, so we start with 5 × 3 = 15 connections. We need a net gain of +5 connections to reach the required 20. Let's look at the net gain of breaking a link: - Breaking an end link destroys 1 existing connection, but the resulting open link can create 2 new connections when resealed. Net gain = +1 connection. - Breaking an internal link destroys 2 existing connections, and the open link creates 2 new connections. Net gain = 0. - Breaking an already isolated link (0 connections) gives 2 new connections. Net gain = +2. To get the maximum net gain, we should break links from a single chain (A-B-C-D) sequentially: 1. Break A (end link): destroys 1 connection, net gain +1. (Remaining: B-C-D) 2. Break B (end link): destroys 1 connection, net gain +1. (Remaining: C-D) 3. Break C (end link): destroys 1 connection, net gain +1. (Remaining: D, which is now isolated) 4. Break D (isolated link): destroys 0 connections, net gain +2. Breaking all 4 links of one chain gives a total net gain of +5 connections (1+1+1+2), taking us perfectly from 15 to 20 connections. If we only make 3 breaks, the absolute maximum net gain we could achieve is +3 connections (1+1+1), which only brings us to 18 connections—two short of a loop. Therefore, it is mathematically impossible to complete the necklace with 3 breaks. We must make 4 breaks at $10 each, totaling $40.

Final Summary

The AI Roundtable converged on a $40 solution to the jeweler's puzzle, with GPT-5.4, Gemini 3.1 Pro, and Grok 4.1 Fast immediately identifying that sacrificing one entire chain was more efficient than a naive one-break-per-chain approach. Claude Opus 4.6 initially caused a stir by opting for the $30 price point, but ultimately recanted after realizing that three links cannot mathematically close a loop of five segments. Gemini 3.1 Pro provided the knockout blow with a rigorous proof based on net connection gains, solidifying the consensus.

All 4 models agreed on "$40" after discussion

Strongest Arguments

• $40: To reach the 20 connections required for a loop from five 4-link chains (which start with 15 internal connections), one must achieve a net gain of +5; breaking all 4 links of a single chain provides exactly this gain (+1 for each of the first three links and +2 for the final isolated link).
• $30: Breaking 3 links from a single 4-link chain to use as connectors, leaving the 4th link as a segment, which theoretically reduces costs if the resulting five segments could be joined with only three breaks.

Key Turning Points

• Gemini 3.1 Pro introduced a mathematical proof showing that a net gain of five connections is required, making $30 (3 breaks) physically impossible.
• Claude Opus 4.6 abandoned its support for Option A after realizing their own logic for three connectors only produced a linear chain rather than a closed loop.