Puzzle Tournament 3

As we say “goodbye” to one of the worst years in recent memory, we at Puzzle Prime are looking forward to a more prosperous and fun 2021. For that reason, we decided to launch the new year with our third puzzle tournament, which will be organized on January 31, Sunday.

This time, we will collaborate with other talented puzzle creators to bring you a large variety of unique and challenging puzzles. To register for the tournament, simply fill the form below.

You can check out our previous puzzle tournaments HERE and HERE.

Topple Magazine

Review

Topple is a new online magazine that can be printed out and solved with pen and paper, the old-fashioned way. The price of each issue is just a meager $1, and for that amount you get a PDF of about 10 pages and up to 20 puzzles.

The authors of the magazine have put efforts to provide a large variety of puzzles, so that everyone will find something they like. There are interesting trivia questions, rebuses, grid logic puzzles, as well as some unique challenges I have never encountered before. The number of puzzles in the magazine keeps increasing with each next issue:

Issue 1: 7
Issue 2: 7
Issue 3: 10

Issue 4: 10
Issue 5: 10
Issue 6: 12

Issue 7: 16
Issue 8: 18
Issue 9: 18

The quality of the puzzles keeps improving as well, as you can see from these two identical puzzles, second one of which has updated artwork.

Before you purchase the magazine, you can take a look at the 4-page sampler we have provided below.

If you enjoy the Topple puzzles as much as we do, we recommend you to get the latest issue and then keep buying the older ones in a reverse order. At just $1, it will be hard to find something that provides better value for the money.

  • appropriate for all ages
  • PDF booklet for easy printing
  • about 10-20 puzzles per issue
  • fun old-school vibes
  • just $1

GET TOPPLE MAGAZINE HERE

* This post contains affiliate links. We may receive a small compensation if you make a purchase.

Brain Drop Podcast

Brain Drop is a new puzzle podcast by Brian Hobbs, released on a (mostly) weekly basis. In each episode, Brian presents 3 new puzzles and shares the solutions of the puzzles from the previous week. He uses professional voicework, music, and sound effects, to set up the mood and make his show more entertaining. Click the banner below to check out Brain Drop and see if you can answer Brian’s latest set of puzzles!

Rapunzel and the Prince

The evil witch has left Rapunzel and the prince in the center of a completely dark, large, square prison room. The room is guarded by four silent monsters in each of its corners.  Rapunzel and the prince need to reach the only escape door located in the center of one of the walls, without getting near the foul beasts. How can they do this, considering they can not see anything and do not know in which direction to go?

The prince must stay in the center of the room and hold Rapunzel’s hair, gradually releasing it. Then, Rapunzel must walk in circles around the prince, until she gets to the walls and finds the escape door.

Is This Prime?

In the past few days, I, my friends, and a whole lot of Twitter people have been trying to beat each other’s scores in the game “Is This Prime?”.

The game itself is simple; you are shown random integers on the screen and you need to guess whether they are prime or composite. Since most presented numbers are between 1 and 200, after a couple of games, players naturally memorize them. However, this is a good opportunity for students to review some main number division rules.

  1. Numbers that end with an even digit are divisible by 2. If the number formed by the last 2 digits of a number is divisible by 4, the original number is also divisible by 4. If the number formed by the last 3 digits of a number is divisible by 8, the original number is also divisible by 8.
    • 536 is divisible by 2 because 6 is an even digit
    • 1348 is divisible by 4 because 48 is divisible by 4
    • 71824 is divisible by 8 because 824 is divisible by 8
  2. Numbers that end with 5 are divisible by 5. Numbers that end with 25, 50, 75, or 00 are divisible by 25.
    • 45 is divisible by 5
    • 675 is divisible by 25
  3. Numbers whose sum of digits is divisible by 3 are divisible by 3. Numbers whose sum of digits is divisible by 9 are divisible by 9.
    • 144 is divisible by 3 because 1+4+4=9 is divisible by 3
    • 1638 is divisible by 9 because 1+6+3+8=18 is divisible by 9
  4. If the difference between the sum of the digits in odd places and the sum of the digits in even places is divisible by 11, the number is divisible by 11.
    • 121 is divisible by 11 because 1+1-2=0 is divisible by 11
    • 209 is divisible by 11 because 2+9-0=11 is divisible by 11
    • 1628 is divisible by 11 because 1+2-6-8=-11 is divisible by 11
  5. If the number before the last digit minus twice the last digit is divisible by 7, the original number is also divisible by 7.
    • 161 is divisible by 7 because 16-2×1=14 is divisible by 7
    • 371 is divisible by 7 because 37-2×1=35 is divisible by 7
    • 1589 is divisible by 7 because 158-2×9=140 is divisible by 7

All the rules above apply in both directions, e.g. if the sum of the digits of a number is not divisible by 9, then the number itself is also not divisible by 9. There are more complicated rules that apply to larger numbers but the chances are you will never get to use them. If you are curious to learn more about them, go to the bottom of this article.

Once we know the main number division rules well, we are ready to play the game! Here are a few tips for getting high scores:

  1. Memorize as many numbers as possible. Knowing the multiplication table up to 10×10, it should be easy to learn by heart whether each number up to 100 is prime or composite.
  2. Pay attention to the last digit. If it is 5, then the number is composite (unless it is =5).
  3. Check whether the sum of the digits is divisible by 3. If it is, then the number is composite (unless it is =3).
  4. If the number is between 100 and 300, check whether the sum of the first and the third digits equals the second digit. If this is true, then the number is divisible by 11, and therefore it is composite. 209 is the only other number in this range divisible by 11.

Good luck playing and let us know if you beat our personal record of 67 points!

Primes between 1 and 300:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Sneaky composites:

51, 57, 87, 91

119, 133, 161, 169

209, 217, 221, 247, 253, 259, 287, 289, 299

Some Math

Some of you may be wondering why the rules listed above work and whether we can create similar rules for larger numbers. Here are some explanations for the curious among you:


Rule for division by 3

Consider the 5-digit number ABCDE. It can be written as:

ABCDE = 10000A + 1000B + 100C + 10D + E

Since 10000 = 3 × 3333 + 1, 1000 = 3 × 333 + 1, 100 = 3 × 33 + 1, and 10 = 3 × 3 + 1, we can see that:

ABCDE = 3 × (3333A + 333B + 33C + 3D) + (A + B + C + D + E)

Therefore, ABCDE is divisible by 3 if and only if (A + B + C + D + E) is divisible by 3.


Rule for division by 11

Consider again the number ABCDE. Since 10000 = 11 × 909 + 1, 1000 = 11 × 91 – 1, 100 = 9 × 1 + 1, and 10 = 11 – 1, we can see that:

ABCDE = 11 × (909A + 91B + 9C + D) + (A – B + C – D + E)

Therefore, ABCDE is divisible by 11 if and only if (A – B + C – D + E) is divisible by 11.


Rule for division by 7

Once again, consider the number ABCDE. Notice that it can be written as:

ABCDE = 10 × ABCD + E

Now, let us find a number X such that 10X gives remainder 1 when divided by 7. Such number is X = 5. Indeed, 5 × 10 = 50 = 7 × 7 + 1. Therefore, the following statements are equivalent:

  • ABCDE = 10 × ABCD + E is divisible by 7
  • 5 × ABCDE = 49 × ABCD + ABCD + 5E is divisible by 7
  • ABCD – 2E is divisible by 7

Rules for division by 13, 17, 19, etc.

The idea of the rule for division by 7 can be applied to rules for divisions by higher numbers. For example, here is how we can find a rule for division by 13:

  1. Find the smallest positive integer X, so that 10X is divisible by 13. Such number is X = 4. Then, ABCDE is divisible by 13 if and only if 4 × ABCDE is divisible by 13.
  2. Rewrite 4 × ABCDE as:
    4 × ABCDE = 39 × ABCD + ABCD + 4E
  3. Conclude that ABCDE is divisible by 13 if and only if ABCD + 4E is divisible by 13.

As an exercise, try to deduce a similar rule for division by 17!


For deeper understanding of how division of integers works, we recommend our more enthusiastic readers to look into Modular Arithmetic.

The Puzzle TOAD

The Puzzle TOAD is a website, created by four Carnegie Melon professors (Tom Bohman, PO Shen-Loh, Alan Frieze, Danny Sleator), where you can find a growing collection of ingenious math brain teasers. Unlike Puzzle Prime, The Puzzle Toad is targeted exclusively towards math and computer science majors. Students who are preparing for college Olympiads will find the problems particularly useful. Check out The Puzzle TOAD by clicking the banner below.

Minecraft: Magnetic Travel Puzzle

Review

Minecraft: Magnetic Travel Puzzle (M:MTP for short) is a travel game by ThinkFun in which the goal is to arrange 3 types of objects, each coming in 3 different colors, in a 3 by 3 grid, such that certain conditions are satisfied.

As you progress through the 40 included challenges, the types of conditions you encounter become gradually more complex. While in the beginning you may be given all the colors of the objects with one clue and all the types of the objects with another, later on you need to analyze 5 or 6 clues at once, which makes the game more challenging and fun. That being said, at the hardest levels, M:MTP is still relatively easy, so experienced puzzlers will probably breeze through it within an hour or two.

At its core, M:MTP is identical to ThinkFun’s previously released Clue Master. Both games are presented in the form of magnetic notebooks, so they are easy to pick up and travel around with. The illustrations of the Minecraft edition are all based on the popular video game, so its fans may be particularly appreciative.

If you are looking for a casual puzzle to pass an hour or two on a road trip, then M:MTP would be a great choice. I only wish there were more challenges included, especially more difficult ones.

  • 1 player, 8 years and up
  • 40 challenges with increasing difficulties
  • easy to transport and play on the go
  • cool Minecraft based art
  • most puzzles can be solved with a few simple techniques

GET M:MTP HERE

* This post contains affiliate links. We may receive a small compensation if you make a purchase.

Perfect Day

Dear Diary,
This morning, I woke up at 9 o’clock. My first job was to drink a glass of water and brush my teeth. After that, I ate an apple, 2 bananas, and a toast with almond butter for breakfast. Then, I went to the gym for one hour, took a shower, and came back home at eleven. I cooked a few pieces of chicken with sweet potatoes for lunch; ate everything. In the afternoon, I started working on my projects while listening to classical music. In the evening, I went for the 2nd time to the gym, did cardio for half-an-hour, and after that visited my friend’s apartment. We had together one big pizza for dinner and in the end, I fell asleep on his couch.

As you can see, my entire day was perfect, except that my laptop keyboard broke. Can you figure out what’s wrong with it?

All of the numbers in the text are written using digit symbols except for the ones containing “one” (“one”, “eleven”, “first”). This inconsistency implies that the number “1” key on the keyboard is broken.

Puzzle Tournament 2

Puzzle Prime’s second puzzle tournament was organized on June 27, 2020. Congrats to Elyot G. who solved all the puzzles! You can see the problems and the rankings below.

Elyot G.

You have 60 minutes to solve 6 puzzles, each worth 1 point. Upload your solutions as a pdf, document, or image, using the form below. Good luck!

Time for work: 1 hour

Puzzle Prime Knight

Start from a square with a P on the chessboard, and keep jumping via knight’s move, consequently landing on squares with the letters U-Z-Z-L-E-P-R-I-M-E.

Point of View

8 of these diagrams correspond to views of the object in the corner when it is looked from different perspectives. Which 2 aren’t?

Note: The projections below are parallel (not perspective).

Chess Fight

Choose a chess piece on the board. Then, move the piece to a cell with another piece, and remove the first piece. Repeat, by moving the second piece to a cell with a new piece, and removing it. Continue until there is only one piece remaining on the board.

Note: For example, we can move the Queen on d2 to the Bishop on b2, the Knight on c3, or the Bishop on d4. If we move the Queen to d4 and remove it from the board, then we must move the Bishop on d4. The only available cell is c3, where a Knight is positioned. We must remove the bishop and move the Knight on c3 either to the Rook on b1 or the Rook on a2…

Special Date

On April 5, 2013 (5.4.2013), the digits used for expressing the date were all different and consecutive. When was the last date before it with this property?

Remark: The digits 9 and 0 are not consecutive.

Splitting the Area

You have 1 square with side length 1 and 2 circles with diameter length 1. Draw a single line so that the resulting areas on the left and on the right of the line are equal.

Notes: You need to specify how you find a line satisfying the condition above.

Chess Connect

The starting and ending positions of 7 chess pieces are shown on the board. Find the trajectories of the pieces, if you know that they do not overlap and completely cover the board.

Notes: The pieces can not backtrack. Two trajectories can intersect diagonally but can not pass through the same square. Only the Knight has a discontinuous trajectory.


Solutions

The answer to Point of View is C and J.
The answer to Special Date is 23.4.1765.
The solutions of the other puzzles are shown below.

Results

PuzzlerP1P2P3P4P5P6Total
Elyot G.1111116
David R.1010114
Rodrigo R.10.510013.5
N1010103
Hristo H.0010102
Rockmosis1010002

Prizes

#1.
$10 Gift Card (iTunes or Play Store)
Puzzle Avatar (custom made)
Puzzle Pelago (mobile game by Hallgrim Games)
Hook (mobile game by Rainbow Train)

#2.
$5 Gift Card (iTunes or Play Store)
Puzzle Avatar (custom made)
Puzzle Pelago (mobile game by Hallgrim Games)
Hook (mobile game by Rainbow Train)

#3.
Puzzle Avatar (custom made)
Puzzle Pelago (mobile game by Hallgrim Games)
Hook (mobile game by Rainbow Train)

#4-5.
Puzzle Pelago (mobile game by Hallgrim Games)
Hook (mobile game by Rainbow Train)