A Maze Puzzle for the Day

Here’s a little maze puzzle I originally built a couple of years ago, that seems apropos to reprise now:

Can you make it from the A in the top left of this grid to the Z in the bottom right, always going either up one letter (for instance, A to B or G to H) or down one letter (for instance, N to M)? The alphabet wraps around, so you can go from Z up to A or A down to Z too. Try as hard as you can (and remember that you can always work backward if you get stuck forwards), and see where you get!

Remark: Solving the maze is not the same thing as solving the puzzle. Read those instructions carefully!

Source: Puzzling StackExchange

Notice this puzzle is published on April 1st. Actually, it doesn’t have a standard solution. If you connect every two consecutive letters which appear next to each other in the grid, you will get two disconnected components, one of which contains the START and the other contains the END. The first component has 5 dead-ends – at letters A, P, R, I, L, and the second component has 5 dead-ends – at letters F, O, O, L, S. These two spell out “April Fools”, which is the real solution of the maze.

The Madman’s Speech

You are walking through the prairie when you find a madman wandering around talking to himself. The following is what you manage to hear of his speech:

“How? I – I’ll ask her. I owe her much, again. I’d a home on town, a tax as florid as out the coat, a virgin a year. Oh, yodel – aware you take all or I do. Never the road: I’ll land in the Anna-Marie land. Main can’s a sore gone; tennis is out t’car. Oh, line a canned turkey!”

What is the man really talking about?

Source: Puzzling StackExchange

The man is actually reciting the states in America, even though you can’t hear him well:

“How? – I” = Hawaii
“I’ll ask her.” = Alaska
“I owe her” = Iowa
“much again” = Michigan
“I’d a ho…” = Idaho
“…me on town, a” = Montana
“tax as” = Texas
“florid a…” = Florida
“…s out the coat, a” = South Dakota
“virgin a year.” = Virginia
“Oh yo…” = Ohio
“…del – aware” = Delaware
“You ta…” = Utah
“…ke all or i do” = Colorado
“Never the” = Nevada
“road: I’ll land” = Rhode Island
“in the Anna-…” = Indiana
“…Marie land” = Maryland
“Main” = Maine
“can’s a s…” = Kansas
“…ore gone;” = Oregon
“tennis i…” = Tennessee
“…s out t’car. Oh line a” = South Carolina
“canned Turkey” = Kentucky

Grasshoppers

Four grasshoppers start at the ends of a square in the plane. Every second one of them jumps over another one and lands on its other side at the same distance. Can the grasshoppers after finitely many jumps end up at the vertices of a bigger square?

The answer is NO. In order to show this, assume they can and consider their reverse movement. Now the grasshoppers start at the vertices of some square, say with unit length sides, and end up at the vertices of a smaller square. Create a lattice in the plane using the starting unit square. It is easy to see that the grasshoppers at all times will land on vertices of this lattice. However, it is easy to see that every square with vertices coinciding with the lattice’s vertices has sides of length at least one. Therefore the assumption is wrong.

Chessboard Madness

You have unlimited number of knights, bishops, rooks and kings. What is the biggest number of pieces (any combination) you can place on a chessboard, so that no piece is attacked by another one?

If we put 32 knights on all black squares, then no two pieces will attack each other. Now let’s see that if we have more than 32 pieces, then there will be two which attack each other. Split the chessboard in 8 rectangular sectors of size 2×4. It is not hard to see that if we have more than 4 pieces in the same 2×4 sector, then 2 of them will attack each other. Therefore we can place at most 4 × 8 = 32 pieces on the chessboard.

Not Twins

A teacher enters the classroom and sees on the first row two students sitting next to each other, looking completely identical. She asks them if they are twins, but the students simultaneously reply that they are not. After checking in the records, the teacher furthermore discovers that the two children have the same mother and father. What is the explanation?

The students are three of a triplet.