The solution to last week’s Microsofty involves a bit of math, so this week we offer a puzzle that challenges you to think outside the box instead: Landscaper Lou was given 10 apple trees and told to plant them in five rows with four trees in each row.

He did it. How?

Solution to Microsofty 93: Square Dance     

Last week’s puzzle got even harder when we had to think in more than two dimensions.

1. We have a 4×4 square of squares, shown below, and we ask ourselves, what is the total number of squares inside the big square? As shown in the picture, we have sixteen 1×1 squares, nine 2×2 squares, four 3×3 squares, and one big 4×4 square. So the result is 1 + 4 + 9 + 16 = 30:

We can see a pattern here:

2. The pattern we see can be extended to n terms, that is, for a square, n × n

the total number of squares is

There turns out to be a closed form solution for this. The total number of squares is equal to

For  n = 4

This is the answer we got before from studying the diagram.

Image Credit: fourtakig - Adobe Stock

3. How many cubes in a 4 × 4 × 4 cube? Careful counting reveals the number:

Again, we see a pattern. This sum is equal to

4. Extending the solution of the previous problem to this one, we have for an n × n × n cube, the formula

There is a closed-form solution for this sum too. It is

For

This is the answer we got from counting the squares.

5. A pattern emerges. For two dimensions, we have the sum of the squares. For three dimensions, we have the sum of the cubes. Therefore, for N dimensions, the total number of hypercubes for a

hypercube (N terms) is

Interestingly, there is a closed form solution for this sum too. (I had to look it up.)

where the   

values are are Bernoulli polynomials. (If you are a nerd, you will enjoy learning more about this math concept!)

The Monday Microsofty is a weekly feature of Mind Matters News. Here are the links to all the puzzles and answers to date:

Monday Microsofty 93: Square dance with the mathematicians! How do we determine how many squares there are in squares of squares? Or cubes of cubes? Hint for the handshake problem in Microsofty 92: Friedrich Gauss (1777–1855) provided a formula for this.

Image Credit: Giulio_Fornasar - Adobe Stock

Monday Microsofty 92: When the nerd enters a room… Shaking hands is expected. Here’s a math question about the custom, to delight the true nerds among us. A hint for Microsofty 91: Where might Jimmy the Mooch have found some spare energy? You can find links and answers to Microsofties 91 through 87 here as well.

Monday Microsofty 87: Are you as good a detective as you think? Here’s a test from the world of codebreaking To solve last week’s puzzle, look up the definition of “calorie.” Are we using the word to mean the same thing? You can find links and answers to Microsofties 77 through 86 here as well.

Monday Microsofty 76: The Smoking Gun explains a computer scientist or engineer, a law enforcement officer often relies on abductive reasoning to crack a case, so with this in mind, you will have to crack last week’s puzzle. You can find links and answers to Microsofties 55 through 75 here as well.

Monday Microsofty 55: “It happens every spring.” Baseball, that is. Here’s a puzzle that takes in baseball’s summer. To solve last week’s puzzle, you don’t need to know the distance. Check the problem again for the number you do need to know. You can find links and answers here to Microsofties 51 through 54 as well.

Image Credit: Oleksiy - Adobe Stock

Monday Microsofty 50: Cutting through the cornbread. How did Yuri Senior cut the cornbread into eight identical portions using only three straight cuts? You can guess the answer to Microsofty 49 if you try the test question yourself at home, using a small mirror. You can find links and answers here to Microsofties 46 through 49 as well.

Monday Microsofty 45: Can Tony beat the fast-food curfew? An early curfew on fast food service motivated a boy to exercise more vigorously. But how fast was he pedalling? To solve Micro Softy 44, recall that Tony doesn’t need to take the individual pills each day, only the prescribed amount of each. You can find links and answers here to Micros Softies 41 through 44 as well.

Monday Microsofty 40: The fate of a false prophet. He wasn’t actually fired for being a false prophet but for something that his prophecy unintentionally revealed. The solution to Micro Softy 39 lies in considering an alternative possible meaning of a word commonly used in sports. You can find links and answers here to Micro Softies 30 through 39 as well.

Monday Microsofty 29: A funeral lament in four lines. The funeral director was puzzled by Dan’s description of his relationship to the deceased but there was no question that his grief was sincere You can find links and answers here to Microsofties 22 through 29 as well.

Monday Microsofty 21: Finding More of the Deadly Fentanyl Pills. You can find links and answers here to Microsofties 11 through 20 as well.

Monday Microsofty 11: What Happened to That Other Dollar? You can find links and answers here to the first ten Micro Softies. Have fun!