# Splitting a Plate into 4 Equal Pieces

You are stuck on an island and have been tasked by the natives with dividing a plate of chocolate into 4 equal pieces, one for each of the island's gods. Each god must have an equal share, or you go into the volcano.

The plate is a perfect square. The only tools you have are a short knife, a perfect circle with diameter equal to the side of the plate, and a Y shaped thingy with angles of 120° and leg lengths equal to the radius of the circle.

You are skilled enough to line up two objects perfectly, and to cut along a marked line perfectly, (but are unable to cut perfectly just eye-balling it). How do you satisfy the island's gods?

• damn you, i want chocholate cake as well – user902383 Nov 3 '14 at 11:59
• Eat the chocolate, leaving none for any of the gods. They all get an equal amount of chocolate! (And if that backfires, it's still a pretty good way to die.) – Thane Brimhall Nov 5 '14 at 17:53
• Does the short knife imply that you can only cut along a marked line and not a line defined by two fixed points? Otherwise, you can just cut along the line defined by opposite vertices. – user80551 Nov 8 '14 at 14:28
• Correct. The knife is short enough that you just can't put it across the plate and cut. And you can't cut a straight line without some sort of guide between two points. – JonTheMon Nov 8 '14 at 16:44 1. Place the circle so that its edge touches two of the square vertices.
2. Align your "Y shaped thingy" so that one vertex touches the side of the square and the others touch the edges of the circle.
3. Use the leg of the thingy that lays out of the circle to mark a line on the square with your knife.
4. Rotate the square and repeat three more times.
5. Cut the square using the marks you made.
• How did you align Y? The sides of Y seem much smaller than that of the square. – RBW Nov 2 '14 at 21:55
• @Marko the three sides of Y are equal, and exactly half the side of the square. And that's also the distance from the circle to the left side of the square. – rsanchez Nov 2 '14 at 22:17
• Okay understood. – RBW Nov 3 '14 at 18:26

Lay the circle directly on top of the square, and cut along the outside of the circle. We now have four pieces on the outside which have 2 sides both equal to half the width of the square, all rotations of this: Take one piece and line it up on the circle such that its two cusps just meet the edge of the circle, and cut: This cuts one quarter from the circle. Align either piece so that one flat side matches with the new flat part of the circle, and the other flat side is a radius. Cut and repeat. We now divided the square into two sets of four equal pieces, not even using the Y shaped tool, four like this: And four like this: • Note that I'm assuming I can use the edge of a piece of chocolate to cut along. – Justin Oct 31 '14 at 22:52
• You can cut along an edge, that's fine. I'm not sure if I like that you have 8 pieces of chocolate, but, well, being chocolate that might be easily remedied. And you can line up the Y shaped piece and continuously cut along that. – JonTheMon Oct 31 '14 at 23:04
• @JonTheMon Can I press lightly into the chocolate to mark it first? – Justin Oct 31 '14 at 23:08
• Sure. As long as it doesn't remove any chocolate. – JonTheMon Oct 31 '14 at 23:15

This method divides the square plate of chocolate into four equal pieces, as shown in the diagram below.

Initial steps:

Align the circle so that a diameter of it lines up with one side of the square (as in the first diagrams of two of the three previous answers). Mark a short arc near the center of the square. (That is, lightly score a small distance. This initial arc can be made arbitrarily short, if desired, by using the three-armed item as in step 2 of rsanchez's answer, and can be so made that it is later cut over.)

Then do the following four times:

Move the circle from its current side-alignment to align with an adjacent side of the square. Cut a quarter arc counterclockwise from a corner of the square, stopping each cut upon meeting the initial arc or a previous cut. • This is beautiful because the pieces aren't composed entirely of lines. – goodguy5 Mar 23 '16 at 19:51

We can divide it into 4 equal pieces without dividing it into 8. Like so:

Line the circle up so that its diameter is over a side of the square. Mark the arc that passes through the square. Do this for the opposite side, resulting in a square marked like so: Use the Y-shaped piece to find the midway point of any side (this will be where all three prongs join), rotate it until one prong touches both circles (center of square). Cut along this line.

Repeat for all sides. We now have four pieces that are rotations of: Note that the arc in the middle didn't actually cut through the chocolate; it's just a mark. Align the circle so that it is tangent to each side of the square. Align the Y thing between the chocolate and circle so that it touches the circle at each tip, as shown. The radius of a circle that touches a point of tangency is perpendicular to that tangent, so the vertical piece of the Y thing marks the position of half of the vertical center line of the chocolate. Rotate it 180° to finish the first bisector, then complete the horizontal division line in a similar way. This will produce 4 equal size squares of chocolate.

In my previous answer, I used only the circle to divide the square plate of chocolate into four equal pieces. In this answer, I show how to divide it equally using only the Y shaped thingy.

The general idea is to use thingy to make four long cuts toward the center, each identical relative to its side of the square; and then make two slices across the short gaps to complete the cutting.

Part 1, Mark midpoints of sides:

On each side of the square, place thingy with one post-end in a corner, and the Y-junction clockwise (of that corner) on the side. [See first diagram.] At each placement, mark the Y-junction location (ie, the midpoint of the side).

Part 2, Make four long cuts toward the center:

On each side of the square, place one thingy post-end at the side's mid-point mark, and another post-end against the counterclockwise adjacent side. [See second diagram.] Cut from the mid-point's post-end up to the Y-junction. [Net result is as shown in third diagram.]

Part 3, Make two slices across the short gaps to complete the cutting:

Make two short cuts across the plate-center to connect the ends of opposite long cuts, as shown in red in the fourth diagram, using any leg of thingy as a straightedge.    Edit: A slight variation of this method allows cutting the large square into four equal small squares (instead of shapes with six edges each) using only the 3-leg thingy. Rather than making the long cuts shown in green in the fourth diagram, just mark their endpoints near the center,and make the red lines as marks also. Their intersection marks the center of the plate. Now use a thingy leg as a straightedge to cut from each side's midpoint to the center.

Cut the circle in half using the plate as a guide. Perfectly align the two pieces edge to edge and use them as a straight edge to cut the cake diagonally, twice.

• The Y-thingy (instead of the plate) can be used to cut the circle in half, if the circle can be cut. (It might well be that a perfect circle cannot be cut or marked.) – James Waldby - jwpat7 Nov 6 '14 at 19:14

Take Y shaped one and mark on four sides of the circle as one leg equals to the radius. you get marking on four edges of the plate. Now mark the center of the plate by keeping the legs touching the circle. You have five markings. Now using legs of Y mark lines joinng the center marking from edges. You have perfect lines and use knife to cut So here's an another way to divide the square into four Equal parts. I hope this will help u :)

• Welcome to Puzzling! Can you explain how the tools in the question will help you make these cuts? – Glorfindel Jul 27 '18 at 8:42