If you like this puzzle,
you might also like...

## detailed solution

We know from Brian's comments how long the day lasts, as well as when lunch occurs.

Brian: "The day lasted from 9 AM to 5 PM. Everyone got done right at noon, so we all broke for lunch for half an hour."

This lunch break is important, because we know from the instructions that no task is left unfinished. That means that everything that is started in the morning is completed before lunch. The sum of all the tasks' person-hours is 45, which also equals 6 people x 7.5 hours. Thus, every person will be occupied with tasks constantly throughout the day, without any gaps.

Before trying to specify when any of the tasks occur, notice that several of your friends comment on the necessary order of certain tasks. These task interdependencies should (hopefully) give us some idea of when the tasks need to be accomplished.

Colleen: "We knew we’d be the last step on a lot of tasks. The trail sign could only be painted once it was rebuilt, same with the park bench and the fence."

Three different painting tasks come after three associated carpentry tasks: park bench repair, trail signage repair, and fence repair.

Troy: "The bridge is at the start of the trail, so that had to be repaired first. Then trail maintenance could be done, and finally that sign could be replaced."

Several trail tasks must come in a certain order, including the unmentioned signage painting.

Morgan: "There was a lot of underbrush that needed to be removed, so we did that first. That got cleaned up during leaf raking and was taken to be mulched on the gardening supply run. Then the flower beds had to be weeded before flower planting, and finally mulch spreading."

Several gardening tasks needed to be accomplished in a particular order as well. Note that leaf raking and mulch spreading are listed as generic tasks that could be done by anyone. (In addition, all the supply runs were also listed as generic tasks.)

Focusing on the trail tasks first, if Troy and Amanda wait to begin the bridge repair until after lunch, even with two people working on each expertise-specific task at a time, the trail sign won't be able to be painted until after 5 PM. This means that the bridge repair must occur in the morning.

Jake: "All the supply runs needed two people. The painters and carpenters couldn't do anything without the donated supplies, so we went first thing."

At 4 person-hours, both Troy and Amanda must complete the bridge repair together in order to finish in the 3 available hours between 9 AM and noon. Plus, since the morning bridge repair would require lumber, the carpentry supply run would need to be done first.

Similarly, if Morgan and Brian waited to remove the invasive underbrush until after lunch, even ignoring the leaf raking and mulch spreading (since it is unclear how many people could help, and thus just how long each would take), the weeding and flower planting would extend past 5 PM. This means that the invasive removal must also occur sometime in the morning. We'll come back to when exactly this should occur.

If one of the gardeners completes the trail maintenance by themselves, the earliest the trail sign painting could be completed is 5:15 PM. Thus the trail maintenance must be completed by both gardeners together.

Likewise, if the weeding is done by only one gardener, flower planting would commence at 5 PM. (Remember that flower planting must be done after the weeding is complete.)

Or if the flower planting is done by only one gardener, that would be completed at 5:30 PM.

Thus each of these expertise-specific tasks must be accomplished by both gardeners working together. If weeding proceeds trail maintenance, we run into the same situation we had when trail maintenance was accomplished by only one person, delaying trail sign repair and paint.

Thus trail maintenance must occur right after lunch. While this work distribution leaves some time at the end of the day, the mulch spreading still needs to be accounted for.

Now let's sort out how the remaining carpentry (and painting) tasks can be accomplished in the afternoon. Recall that there are three sets of tasks that need to be accomplished in sequence.

Amanda: I repaired the fence when we split up.

This indicates that at least some of the carpentry tasks are accomplished individually. If Troy and Amanda repair the trail sign together as soon as the trail maintenance is complete, Amanda won’t have enough time between lunch and 2 PM to finish the fence repair on her own, nor will she be able to finish before 5 PM after the sign repair is complete.

Amanda could begin the fence repair right after lunch. Once that's complete, Amanda and Troy could repair the sign together.

And this still leaves enough time for the other two carpentry tasks in the afternoon. The park bench repair should be completed as early as possible (by Troy individually), since the bench needs to be painted later, and the playground equipment does not. (This leaves a conspicuous hour gap for Troy, but we’ll worry about that later.)

Now we need to make sure that all the painting can be accomplished after each respective carpentry task. If Jake and Colleen try to complete each task separately as soon as possible, both the sign and fence painting run past 5 PM.

However, splitting them up really does not make a lot of sense, especially if one notices that the two person-hours of painting the park bench can be completed by both painters working together before the fence repair is completed at 3 PM. This suggests that the painters should work together on each task. If they do, all three painting tasks can be completed by 5 PM.

However, mulch spreading needs 2 person-hours and must be accomplished after flower planting. In the current arrangement, the only people available in the late afternoon are the gardeners, and they cannot complete the task before 5 PM.

Mulch spreading exceeds the 5 PM deadline by just 1 person-hour. Since neither the painting nor the gardening tasks can be moved back, it's time to address the problematic hour gap for Troy. If we can push up the times for the late-day carpentry repairs, that hour of available time will trickle down to the end of the day, where the carpenters could help the gardeners spread the mulch.

If we try to move up the playground equipment repair by having Troy complete that task by himself, it delays the trail sign repair, which causes a subsequent delay in the painting of the trail sign until 4:45 PM. In addition, for the carpenters to help the gardeners spread the mulch, that too would need to wait until 4:45 PM. All of these tasks would then be completed at 5:15 PM.

Instead, let Troy start work on the trail sign at 2 PM. When Amanda finishes the fence repair, she can complete the playground equipment repair by herself. Both of these tasks would be complete at 4:30, just when the gardeners are finishing the flower planting, so that all of them can spread mulch until 5 PM. This also leaves enough time for Jake and Colleen to paint the sign.

Jake: The gardeners could start right away [without completing the supply run]. They just needed the flowers and mulch to be picked up at some point.

In the morning, the gardeners must work together to complete the 4 person-hours of invasive species removal in the 3 available hours. But then 4 person-hours of leaf raking must also be accomplished after the underbrush removal but before the gardening supply run. The only way all of this can be completed is if the painters and gardeners work together to complete the leaf raking. And, as Jake indicates, the gardeners can begin the day with the underbrush removal before the gardening supply run is completed.

Since Jake and Colleen are the only people to have available time, they must be the ones to go on the gardening supply run. Because they need to take the removed underbrush to be mulched, the supply run cannot happen in the morning. Thus the gardening supply run must be completed in the afternoon, which is still before the gardeners need the flowers and mulch.

Because the graffiti elimination will need paint supplies, the paint supply run must be completed first thing in the morning.