Connect every island: when the cheapest line is a trap

Before you start

Use addition and compare whole numbers. Start on paper without a timer. Letters name islands; a cost is a puzzle number, not a real energy measurement.

Work through this example

Connect A, B, C and D using exactly three of the six lines in the table. You can travel along each chosen line in either direction. Every island must be reachable from every other, and the total cost must be as small as possible.

Reason step by step

1. Take A–B for 1, then B–C for 2. A, B and C are now one connected group. D is still alone.
2. Skip A–C for 3. A already reaches C through B, so this line would close a loop without bringing D into the group.
3. Take C–D for 4. All four islands connect, using three lines. The total is 1 + 2 + 4 = 7.
4. Why is 7 the minimum here? Any solution needs a line to D costing at least 4. Its other two distinct lines cost at least 1 and 2. That gives a lower bound of 7, achieved by this network.

A mistake worth understanding

Choosing the three cheapest lines costs only 6 but leaves D isolated. Connecting everything is essential. A–B, B–C and B–D connects all islands for 9, but it is not the cheapest network.

Try it yourself

Keep every cost except B–D, which now costs 3 instead of 6. Which three lines would you choose? Explain why keeping A–C for 3 still does not help.

Use the idea in the game

Open Island Grid in Logic Lab. Level 1 has five islands, level 2 has six; connect them with four or five lines respectively. Compare the printed costs, not how short lines look. Before submitting, trace a route to every island and check for loops.

About & help

These examples were written for Piko Game. Generated game questions may differ. A score describes a game attempt, not mastery of a school curriculum.

Content and corrections

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