14
$\begingroup$

Woody Woodcutter is preparing his special slanted wooden blocks box. His nephew likes to put slanted roofs on every construction, so he made a special box with many slanted blocks to add to his collection.

Here is what he intended to build.

a box 7x7 of nicely arranged slanted wooden blocks

These are 1x1x3 and 1x1x2 wooden blocks, most of them cut in halves along the diagonal. All packed in a nicely crafted 7x7 box.

But as he was painting the blocks, he accidentally dropped one of them in the box before it was dry, and now that it is dry, it is stuck to the bottom of the box. No way to detach it.

An empty box with a box with coordinates (4.00, 4.00) (4.28, 3.04)(2.36, 2.48) (2.08, 3.44)

He has no time to build a new box. So he has no other option than to squeeze as many blocks as he can inside the box, and that will be it.

Your task, should you accept it, is to fit as many blocks as possible inside the box around the already placed block.

The set of blocks to place is shown in the first picture. Aim for the maximum area covered. The corners of the "randomly placed" block are at (4.00, 4.00) (4.28, 3.04) (2.36, 2.48) (2.08, 3.44).

Should I mention it? No "outside of the box" solution is expected.

Hint:

The initial block placement is not random at all. And Weather Vane asked a good question.

Improve this question
$\endgroup$
6
  • $\begingroup$ Is it correct to assume the the bottom row of blocks are all full (no diagonal cuts), and that the block stuck in the box is one of these? $\endgroup$ Commented yesterday
  • $\begingroup$ Can they all be placed? (Was the block randomly placed?) $\endgroup$ Commented yesterday
  • $\begingroup$ @AxiomaticSystem This is correct. $\endgroup$ Commented yesterday
  • 2
    $\begingroup$ Aside: Is the last name supposed to be Woodcutter, or are they just a cuter version of wood? :P $\endgroup$ Commented yesterday
  • 2
    $\begingroup$ Woodcutter it is. What is cute is the mistake. $\endgroup$ Commented yesterday

3 Answers 3

Reset to default
14
$\begingroup$

There is an exact solution that fits all parts:

Solution

With overlaid grid:

SolutionWithLines

Edit: The idea behind it:

The exact coordinates made me suspicious at first (as well as the fact that the description sounded like an exact solution is possible). I noticed that the offset isn't random but a 7-24-25-pythagorean triple. This reminded me of a fact I came across when watching a video from a lego builder, who used a technique he called "sugar grids" to offset the 90°-stud direction to another angle, essentially using kite shapes to rotate around angles described by such pythagorean triples. I also noticed that by combining two kites made of 2-by-1 and 3-by-1 triangles, it wasn't too hard to arrive at the exact angle of the rotated piece. I got stuck a bit at first because I initially thought that exactly one kite of each type would be needed to arrive at that precise angle (with their tips in opposite directions), but later found out that combining kites of the same type can also give me the correct angle. From there on, it was just a bit of trying to align to the edges with different kite shapes, and then filling the rest with normal 1x2-pieces. Great puzzle!

Improve this answer
$\endgroup$
2
  • $\begingroup$ Very nice. How did you find it? $\endgroup$ Commented yesterday
  • $\begingroup$ @quarague The explanation was too long for a comment, see the edit in my post. $\endgroup$ Commented yesterday
4
$\begingroup$

I can recover

47.5

out of 49 unit squares:

The picture only shows a 4x3 subgrid at offset x=2,y=0.

blocks of wood

Improve this answer
$\endgroup$
2
  • $\begingroup$ Rot13(V cerfhzr vg jnf vagragvbany gung gur frrq oybpx vfa'g ng dhvgr gur fnzr natyr nf gur ynetre jrqtrf, abg rira jvguva gur gjb qrpvzny cynprf hfrq.) $\endgroup$ Commented yesterday
  • $\begingroup$ @WeatherVane Would be too easy, maybe? $\endgroup$ Commented yesterday
2
$\begingroup$

Here’s my attempt (sorry for flipping all blocks about the vertical):

my attempt at filling the box with all blocks except a 2×1 block

This arrangement covers

all blocks except a 2×1 block, so covers an area of 47 out of 49.

Improve this answer
$\endgroup$

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.