A Number Trick Puzzle

A. Think of any number.

B. Multiply your number by 6.

C. Add X.

D. Divide by Y.

E. Subtract the number that you initially thought of.

Your answer is 7.

What values can you give X and Y so the above is always true?

Scroll down for a clue and further down for the answer.

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Clue: You can solve this puzzle using algebra or trial and error. If you are using algebra, let n= your initial number and put everything in the question into an equation that equals 7.

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Answer: X = 42 and Y = 6.

This means:

A. Think of any number.

B. Multiply your number by 6.

C. Add 42.

D. Divide by 6.

E. Subtract the number that you initially thought of.

Your answer is 7.

MOM Squared

Each of the letters in the above equation represents a different digit from 1 to 9 so that it is a balanced equation. What digit does each letter represent?

Scroll down for a clue and further down for the answer.

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Clue: Focus mainly on the values of the M and O within ‘MOM’. You can narrow down possible digit values for these letters when you notice what happens when you get an answer by squaring ‘MOM’.

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Answer: (363)^2 = 131,769

This means:

A=1

M=3

N=9

O=6

Z=7

Partitions of 100

Today’s puzzle is about forming numbers as the sum of positive integers.

The number 2 can only be formed by 1 + 1.

The number 3 can be formed in two ways: 1 + 1 + 1 and 2 + 1 (1 + 2 is considered the same as 2 + 1).

The number 4 can be formed in four ways: 1 + 1 + 1 + 1, 2 + 2, 3 + 1 and 2 + 1 + 1.

The number 5 can be formed in six ways: 1 + 1 + 1 + 1 + 1, 2 + 1 + 1 + 1, 2 + 2 + 1, 3 + 1 + 1, 3 + 2 and 4 + 1.

Your challenge today is to estimate how many different ways the number 100 can be formed as the sum of positive integers.

Scroll down for a clue and further down for the answer.

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Clue: The number is much higher than most people estimate.

Scroll down for the answer.

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Answer: There are 190,569,292 ways to form 100 as the sum of positive integers.

In maths, a partition is an area of number theory that describes the number of ways of writing ‘n’ as the sum of positive integers.

To work out the exact answer for the number of partitions of 100, a computer is required. However, the famous mathematician, Ramanujan, came up with an approximation function to determine the number of partitions of a number. This formula appears below.

By replacing ‘n’ with 100 in the above formula, we get the answer 200 million- a figure relatively close to the actual answer of 190,569,292.

Beat a Calculator With This DIY Maths Trick

1. Cut out four pieces of paper and write the numbers on them that appear in the image below.

2. Fold the four pieces of paper so each one appears like a rectangular prism. Sticky tape can be used to join the edges.

3. Ask a friend to arrange the 4 columns side-by-side. The columns can appear in any order and can be rotated in any orientation.

4. Four 4-digit numbers have been created, reading across. In the above image these numbers are:

5637,

6482,

8246,

7879.

5. You will challenge your friend to add the numbers up as fast as possible however your friend can use a calculator. In less than 1 second you can write down the sum of the numbers. In this instance it is 28,244.

The Secret

You are not really adding the numbers up but instead using a short-cut that works regardless of how the four number blocks are ordered or orientated. You simply look at the third row. In the example given this was 8246. This is the number you are going to manipulate to provide the answer. You need to write the number 2 in front of this number and minus 2 from the total. This means writing 2 before 8246 becomes 28246. Subtracting 2 from this number results in 28,244. This number is the sum of the four lots of 4-digit numbers in this instance.

This method will always work, just look at the third row number, write 2 before the number and minus 2.

A Unique Driving Puzzle

Alice and Ben drive from Northville to Southville. Alice drives for the first 40km and Ben drives the remainder of the distance to Southville. Alice and Ben then drive back to Northville by the same route with Alive driving the first part and Ben driving the last 50km. Did Alice or Ben drive the furthest and by what distance?

Scroll down for a clue and further down down for the answer.

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Clue: You won’t know how far the two places are apart so you can make up a distance and see how far Alice and Ben drive for that distance. Repeat this with another made up distance.

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Answer: Ben drives 20km more than Alice.

The lowest possible distance between Northville and Southville is 50km. In this instance, Alice drives a total of 40km (40km there and 0km back) and Ben drives a total of 60km (10km there and 50km back). If the distance between Northville and Southville is greater than 40km, then Alice and Ben divide the extra kilometres between them with Ben driving south and Alice driving north. For example, if the distance between Northville and Southville is 100km, then Alice drives 40km there (this number will always be the same for Alice) and 50km back for a total of 90km. In contrast, Ben will drive 60km there and 50km back (this number will always be the same for Ben) for a total of 110km. Without knowing the distance between the locations, Ben will always drive 20km more than Alice.

Appear Like a Maths Genius with this Simple Trick

Ask someone to name a three digit number. It’s better if all of the digits are different. Let’s say they name 146.

You then name a three digit number, 853.

Finally, you ask someone to name another three digit number, say 427.

You immediately solve:

146               853

x427             x427

……. plus  …….

 

The other person can try solving the question with a calculator but you will write down the answer 426,573 instantaneously and inevitably beat them.

The trick: The other person genuinely names any three digit number eg 146 (we’ll call this ‘A’).

The three digit number you name isn’t random, despite you acting like it is random. Your three digit number is the number that is required to add onto their number to reach 999. As they named 146, you need to name 853. This is because 146 (their number) + 853 (your number) = 999. The quick way to determine your number is to work out how to reach 9 for each digit of their number. Eg if they name 123, you write 876. (We’ll call this number that you name ‘B’).

Finally, they name a random three digit number, say 427. (We’ll call this ‘C’).

The question has the form

A                B

xC            xC

……. plus  …….

In the case of our example, we have written down these numbers visibly for everyone to see.

146               853

x427             x427

…….   plus  …….

The shortcut to working out the answer is to write down ‘C’ – 1. In our example, this will be 427-1= 426. This will be the first 3 digits of the answer. The last 3 digits of the answer are determined by the first 3 digits of the answer. We simply work out what number makes them add up to 999. In the case of 426, it will be 573.

This means the answer to the question is: 426,573.

Can you work out:

358          641

x869        x869

……. plus  …….

 

Scroll down for the answer.

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Answer: 868,131

What Number Do You Name?

You and I are going to play a number game. You will go first and name a number between 1 and 10 (inclusive). It will then be my turn and I will add a number between 1 and 10 (inclusive) to your number and I will announce the answer. You will then add a number between 1 and 10 (inclusive) to my answer and you will announce the new total. This process will continue. The person who reaches the number 100 first will be the winner. What number do you announce first?

Scroll down for a clue and further down for the answer.

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Clue: Work backwards to determine the numbers that you want to land on. The penultimate number you want to land on is 89.

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Answer: 1.

It is important to determine winning positions in this variant of ‘Nim’ game. If I announce any number from 90 to 99, you will win. This means you want to announce 89 to force me to announce a number between 90 and 99. 78 is also a winning number as regardless of what number I announce, you can reach 89. By working backwards, the winning numbers are:

89, 78, 67, 56, 45, 34, 23, 12, 1. This means as you are going first, you will win by announcing number 1 and following the optimal strategy. This optimal strategy can be expressed as

You announce 1,

Whatever number I add (X), you add 11-X. This process will continue until you announce 100 and win.

A Magic Square Magic Trick

The effect: The spectator is merely thinking of a number between 22 and 99. You draw a 4×4 magic square and show it to the spectator. The spectator is unimpressed. You reveal to the spectator that the sum of the numbers in row 1 is their secret number. In fact, the sum of the numbers in row 2 is their number, same with row 3 and 4. The same for each column, the same for the diagonals, indeed there are 28 unique combinations on your magic square that reveal their secret number.

The method- The secret Number: There are several ways to present this effect. One method involves asking the spectator to name any number between 22 and 99. You then draw the magic square and appear like a maths whiz. There exist other methods to present the effect in which the spectator doesn’t name a number and from their perspective, they are thinking of a freely selected number that you are supposedly not aware of however gimmicks are normally required for this ends. Another method involves forcing a number on the spectator. This can be achieved by forcing the top two cards from a deck of cards onto the spectator eg using a riffle force or the very simple cross-cut force that requires no skill and can be learnt here: https://www.youtube.com/watch?v=Tkte5Ubo9yc.

The method- The magic square: Regardless of what number the spectator chooses between 22 and 99, there will be 12 of the 16 squares in the magic square that will be the same every time. The remaining 4 squares will depend on the number the spectator has chosen. Fortunately, there is an easy method to remember the numbering of the squares.

1.Firstly, write out the multiples of 18. This covers: 18, 36, 54, 72. Make sure only one digit appears in each square and the numbers are vertical. The location of these digits appears below.

2. Write out the numbers 9, 10, 11, 12. The number 9 appears in the bottom right corner and the numbers increase by 1 along the zig-zag. See the below image.

3. There will be 4 spaces remaining. You must now fill in square ‘A’. A is the spectators number minus 21. Move anti-clockwise around the blank squares and add 1 each time. See the below image.

An example of a completed magic square appears below. This magic square was drawn with the number 46 being chosen as the spectator’s number.

Think of a Number…

A.Think of a number between 1 and 50. Both digits must be odd and both digits must be different. Write the number down now.

B.Think of a number between 50 and 100. Both digits must be even and both digits must be different. Write the number down now.

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A.The most commonly selected number is 37.

B.The most commonly selected number is 68.

In question ‘A’, there are only eight numbers that satisfy all the conditions: 13, 15, 17, 19, 31, 35, 37 and 39. Most people exclude the teen numbers as when asked for an odd number, 1 is commonly overlooked and thus ruled out. 3 and 7 are most common digits selected between 1 and 10 and when placed side-by-side they create the super common ‘37’.

In question ‘B’, there are only six numbers that satisfy all the conditions: 62, 64, 68, 82, 84 and 86. This is surprisingly few considering the initial 50-100 range. People tend to pick numbers that consist of ascending digits (the second digit is larger than the first). The sixties are the first valid range of numbers and this may be why 68 is such a common answer.

When asked to pick a card in the deck, the queen of hearts and ace of spades are by far the most common choices.

When asked to pick a number between 1 and 4, 3 is the most common choice.

When asked to pick a number between 1 and 10, 3 and 7 are the most common selections.

When asked to pick a number between 1 and 20, 17 is the most common number people tend to pick however this number is only selected about 18% of the time.

All of this information can be used to perform mentalism magic tricks in which you may have an envelope. On one side of the paper may be the number 3 and the other side may have the number 7. Depending on if the spectator names 3 or 7 when asked for a number between 1 and 10, show the corresponding side. If they select a different number, then ask them for a suit and remove that card from the deck and perform a different trick!

Removing Pebbles

You are in a room facing your opponent with a table separating both of you. There are 11 pebbles on the table and you and your opponent will take it in turns to remove pebbles from the table. On each turn, the player is allowed to take 1, 2 or 3 pebbles. The person who takes the last pebble loses. It is your turn to take pebbles first. How many pebbles do you take?

Scroll down for a clue and further down for the answer.

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Clue: You are guaranteed to win if a certain number of pebbles remain after your turn. Eventually, you want 1 pebble to remain after your turn. Can you work out what the other numbers of pebbles you wish to remain on previous turns?

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Answer: You take 2 pebbles away so there are 9 remaining on the table after your first turn. If your opponent then:

takes 1, you take 3

takes 2, you take 2

takes 3, you take 1.

5 pebbles will then be remaining on the table with your opponent to move.

If your opponent then:

takes 1, you take 3

takes 2, you take 2

takes 3, you take 1.

After your turn, there will be 1 pebble remaining on the table and it will be your opponents turn and they have to take this pebble and lose.

 

This puzzle is based on the ‘game of nim’. There are variants to the above game in which:

A. The person who takes the last item wins

B. There are more or less items than the 11 mentioned in the puzzle.

C. Players are permitted to take a different range of items than the 1-3 listed in the puzzle.

D. There are many variants online involving rows of items, with only items from one row allowed to be taken per turn.

E. There are many wildly more creative versions of the game including: the 100 game, circular nim, Greedy nim and Grundy’s game.

 

In the puzzle variant, there are certain ‘safe’ numbers that you want to achieve after your turn. These are:

1, 5, 9, 13, 17, 21 etc depending on what number you start at. The trick is to reach the next safe level after each of your turns.

The game of nim doesn’t need to be played with items and you can just as easily annoy your friends by playing this game verbally and beating them.

In the variant during which players can take 1 or 2 items, the safe numbers are:

1, 4, 7, 10, 13, 16, 19.

The formula for determining the safe numbers is:

M= Maximum number of items you are allowed to take in the variant.

N=1, 2, 3 etc. You put these numbers in the equation to determine the safe number for different levels.

1+ (N x (M+1))

Play the game of nim online here: https://education.jlab.org/nim/s_gamepage.html In this variant, the person who takes the last pebble wins. Can you work out the safe numbers?