Warning: This is a problem I have not yet found an optimal (or even very good) solution to. But it seems awfully non-trivial, and not in a genre that I've seen before, so I'm throwing it out there for the Den. ( @bonanova that means you should take a look.)

You have a pet mouse. It's an awfully cute mouse. Kind of like those mice on Pinky and the Brain. And you'd like to make it even cuter by teaching it some tricks. The only problem is that your mouse is, well... let's just say it has the brain of a mouse so it's kind of hard to teach it any tricks. But it is very good at eating food. In fact, it will always manage to find the closest morsel of food and go eat it, then find the closest from its new position and go eat that, etc. until it eats everything in sight. So, you'd like to "teach" your mouse to do some tricks given that behavior.

The first exercise is to place morsels of food at the corners of a square and place breadcrumbs so the mouse runs to diagonal corners as much as possible. In other words, suppose you have a square with points A, B, C, D in clockwise order with the mouse starting at point A and you'd like to make the mouse go to point C (diagonal from point A), then point B (one of the two remaining), then point D (diagonal from point B). There are already some breadcrumbs at points B, C, and D (since there need to be breadcrumbs there if you want the mouse to stop at those points) and your goal is to place breadcrumbs to make the mouse go from A to C to B to D while placing as few breadcrumbs as possible.

That first exercise might be a challenge for your kid brother, but for a BrainDen level challenge (the one I haven't convinced myself I've found an optimal solution for yet): Instead of a square, suppose you have a regular pentagon with points A, B, C, D, and E in clockwise order and want to get the mouse to go from A to C to E to B to D?

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Warning: This is a problem I have not yet found an optimal (or even very good) solution to. But it seems awfully non-trivial, and not in a genre that I've seen before, so I'm throwing it out there for the Den. ( @bonanova that means you should take a look.)

You have a pet mouse. It's an awfully cute mouse. Kind of like those mice on Pinky and the Brain. And you'd like to make it even cuter by teaching it some tricks. The only problem is that your mouse is, well... let's just say it has the brain of a mouse so it's kind of hard to teach it any tricks. But it is very good at eating food. In fact, it will always manage to find the closest morsel of food and go eat it, then find the closest from its new position and go eat that, etc. until it eats everything in sight. So, you'd like to "teach" your mouse to do some tricks given that behavior.

The first exercise is to place morsels of food at the corners of a square and place breadcrumbs so the mouse runs to diagonal corners as much as possible. In other words, suppose you have a square with points A, B, C, D in clockwise order with the mouse starting at point A and you'd like to make the mouse go to point C (diagonal from point A), then point B (one of the two remaining), then point D (diagonal from point B). There are already some breadcrumbs at points B, C, and D (since there need to be breadcrumbs there if you want the mouse to stop at those points) and your goal is to place breadcrumbs to make the mouse go from A to C to B to D while placing as few breadcrumbs as possible.

That first exercise might be a challenge for your kid brother, but for a BrainDen level challenge (the one I haven't convinced myself I've found an optimal solution for yet): Instead of a square, suppose you have a regular pentagon with points A, B, C, D, and E in clockwise order and want to get the mouse to go from A to C to E to B to D?

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