1 to 8 on a Cube
Place the numbers from 1 to 8 on the vertices of a cube so that the sum of the four numbers on every face is the same.
SOLUTION
The solution is shown below:
Category: Brain Teasers
A collection of Math, Chess, Detective, Lateral, Insight, Science, Practical, and Deduction puzzles, carefully curated by Puzzle Prime.
We do not know where this puzzle originated from. If you have any information, please let us know via email.
Sequence 1, 3, 7, 12
Which is the next number in the following sequence:
1, 3, 7, 12, 18, 26, 35, 45, 56, ?
SOLUTION
This is the so called Hofstadter Figure-Figure Sequence.
The sequence of the differences between the consecutive numbers in the original sequence is 2, 4, 5, 6, 8, 9, 10, 11… These are exactly the natural numbers missing from the original sequence. Therefore, the next number should be 56 + 13 = 69.
A Cardinal Points
A cardinal points and says, “thorn, shout, seat, and stew.” Can you explain?
SOLUTION
The sentence is a play on words and anagrams. Thorn, shout, seat, and stew are anagrams of the four cardinal directions of the compass: north, south, east, and west.
One Hundred Rooms
There are 100 rooms in a row in a building and inside each room there is a lamp that is turned off. One person enters each room and switches the lamp inside. Then, a second person enters every second room (2, 4, 6, etc.) and switches the lamp inside. A third person switches the lamp in every third room and so on and so far, until person #100 switches the lamp in room 100. How many lamps are turned on at the end?
SOLUTION
We can see that the only switches that have been switched an odd number of times are the ones in rooms with perfect square numbers.
Indeed, if person N has switched the switch in room M, then person M/N has done that as well. Since person N and M/N coincide only when M=N², the claim above follows.
We conclude that the number of lamps that are turned at the end is equal to the number of perfect squares less than or equal to 100; that is exactly 10 rooms.
Conflicting Words
What unique feature do the following words share?
FRIEND, FEAST, THERE, THOROUGH, FLIGHT, WONDERFUL, RESIGN, ENDURING, PEST, COVERT
SOLUTION
Each of these words contains its antonym as a sub-word:
FRIEND – FIEND, FEAST – FAST, THERE – HERE, THOROUGH – ROUGH, FLIGHT – FIGHT, WONDERFUL – WOEFUL, RESIGN – REIGN, ENDURING – ENDING, PEST – PET, COVERT – OVERT
Annihilating Matrix
The numbers 1, 2, … , 100 are arranged in a 10×10 table in increasing order, row by row and column by column, as shown below. The signs of 50 of these numbers are flipped, such that each row and each column have exactly 5 positive and 5 negative numbers. Prove that the sum of all numbers in the resulting table is equal to 0.
SOLUTION
Represent the initial table as the sum of the following two tables:
Since the sum of the numbers in each row of the first table is equal to 0 and the sum of the numbers in each column of the second table is equal to 0, it follows that the sum of all numbers in both tables is equal to 0 as well.
Source:
Quantum Magazine, November-December 1991
Math Homework
A student was given math homework consisting of the following three problems. What is so special about this homework?
- What would the value of 190 in hexadecimal be?
- Twenty-nine is a prime example of what kind of number?
- At time t = 0, water begins pouring into an empty tank so that the volume of water is changing at a rate V’(t)=sec²(t). For time t = k, where 0 < k < π/2, determine the amount of water in the tank.
SOLUTION
The answer to each of the problems in the homework is hidden within the question itself:
- What would the value of 190 in hexadecimal be?
The answer is be. - Twenty-nine is a prime example of what kind of number?
The answer is prime. - At time t = 0, water begins pouring into an empty tank so that the volume of water is changing at a rate V’(t)=sec²(t). For time t = k, where 0 < k < π / 2, determine the amount of water in the tank.
The answer is tank.
Source:
A Dog Tied to a Tree
A dog was tied to a tree with a 7-meter rope. How did the dog manage to get to its bowl of food which was 15 meters away from the dog?
SOLUTION
The dog was initially on the other side of the tree and the tree’s diameter was 1 meter long. Thus, when the dog moved to the opposite side, it got 2×7+1=15 meters closer to its food.
A Very Cool Number
Find a number containing every digit 0-9 exactly once, such that for every 1≤N≤10, the leftmost N digits comprise a number, divisible by N.
SOLUTION
Let the number be ABCDEFGHIJ.
- The digit J must be 0, so that the number is divisible by 10.
- The digit E must be 5, so that the number comprised of the first 5 digits is divisible by 5.
- The digits B, D, F, H, J must be even, and therefore the digits A, C, E, G, I must be odd.
- The numbers CD and GH must be divisible by 4 and the number FGH must be divisible by 8. Since C and G are odd, D and H must be 2 and 6, or vice-versa.
- Since ABC, ABCDEF, and ABCDEFGHI are all divisible by 3, we see that A+B+C, D+E+F, and G+H+I are divisible by 3 as well. Therefore, D+F+5 is divisible by 3 and since D and F are even, we have either D=2, H=6, F=8, B=4 or D=6, H=2, F=4, B=8.
- D=2, H=6, F=8, B=4. Since FGH is divisible by 8 and G is odd, we have G=1 or G=9. Also, G+I+6 is divisible by 3 and since I is odd, we have G=1, I=3. The only possibilities for ABCDEFG are 7492581630 and 9472581630, but they are not divisible by 7.
- D=6, H=2, F=4, B=8. Since FGH is divisible by 8 and G is odd, we have G=3 or G=7. Also, G+I+2 is divisible by 3 and since I is odd, we have G=3, I=1 or G=3, I=7 or G=7, I=3 or G=7, I=9. The only possibilities for ABCDEFG are 9876543, 7896543, 1896543, 9816543, 1896547, 9816547, 1836547, 3816547. Out of these, only 3816547 is divisible by 7.
We conclude that the only solution is 3816547290.
No Three Pawns on a Line
How many pawns can you place at most on a chess board so that no three pawns lie on a single line, horizontal, vertical, or diagonal?
SOLUTION
Since there are 8 rows on the chess board, if you place more than 16 pawns, then there will be at least 3 that lie on a single horizontal line. An example with 16 pawns is shown below:
