Challenge 4 · mathematics · pre-university

The camera lost the labels

Eight maintenance beacons stand along a test railway. Their IDs and positions are: beacon 1 at 2 km, beacon 2 at 7 km, beacon 3 at 13 km, beacon 4 at 18 km, beacon 5 at 24 km, beacon 6 at 31 km, beacon 7 at 35 km, and beacon 8 at 47 km. Only three beacons were armed during a night run. The train travelled in the direction of increasing kilometre marks. Its camera recorded the three flashes, but a blackout erased their labels. The odometer shows 17 km from the first flash to the second, then 11 km from the second flash to the third. Which three beacon IDs flashed? Enter their IDs in travel order as one three-digit code.

This is a project-original problem with a checked answer and a declared solution-family counting rule.

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Checked result

Reveal the independently checked answer

257 code

2 checked solution families

Scan the first gap, then extend

Generate every 17 km candidate pair and retain only the pair whose second position has an 11 km successor.

The 17 km pairs are 7 to 24 and 18 to 35.
Only 7 to 24 can continue another 11 km, because 35 is listed while 46 is not.
The positions 7, 24, and 35 map to IDs 2, 5, and 7: code 257.

Find endpoints, then reconstruct the middle

Add the ordered gaps to obtain the endpoint span, identify its unique pair, and place the middle relative to either endpoint.

The endpoint span is 17 + 11 = 28 km.
Only 7 and 35 among the listed positions are 28 km apart.
The middle is at 7 + 17 = 24 km, so the IDs are 2, 5, and 7: code 257.

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