100+ Challenging math riddles to keep your mind sharp

Answer: Two dozen (24) 

Explanation :  

Certainly! Let's break down the explanation in a mathematical manner.

The original statement says, "If a hen and a half lay an egg and a half in a day and a half, how many eggs will half a dozen hens lay in half a dozen days?"

We've already established that one hen lays one egg in one day. To represent this, we can use the following equation:

1 hen = 1 egg/day

Now, let's introduce the concept of time and the number of hens increasing four-fold:

Number of hens increased four-fold: 1.5 hens * 4 = 6 hens
Time available increased four-fold: 1.5 days * 4 = 6 days
With these increases, we want to find out how many eggs the 6 hens will lay in 6 days. To do this, we can set up a proportion based on the original information:

(1.5 hens / 1.5 days) = (6 hens / 6 days)

Now, to determine the number of eggs laid by the 6 hens in 6 days, we can calculate it as follows:

6 hens/day * 6 days = 36 eggs

So, when we increase both the number of hens and the amount of time available four-fold, the number of eggs increases 16 times:

16 * 1.5 (original rate) = 24 eggs

This equation illustrates how the number of eggs increases when both the number of hens and the time available are increased four-fold.

Answer: 5 and 7. 

Explanation :  

We have two conditions:

The product of two numbers added to the sum of their squares is 109, which can be represented as:
xy + x² + y² = 109.

The difference of their squares is 24, which can be represented as:
x² - y² = 24.

Now, let's correctly solve these equations:

From Equation 2 (x² - y² = 24), you found that x² = 49, which implies x = 7. This part is correct.

However, the step where we substitute x = 7 into Equation 1 is where the error occurred:

(7)y + (7)² + y² = 109
7y + 49 + y² = 109

Now, let's correct the calculation:

7y + 49 + y² = 109

To isolate the terms involving y, subtract 49 from both sides:

7y + y² = 109 - 49

Now, it should be:

7y + y² = 60

Next, rearrange the terms:

y² + 7y = 60

To solve for y, let's rewrite the equation in the form of a quadratic equation:

y² + 7y - 60 = 0

Now, you can factor the quadratic equation:

(y + 12)(y - 5) = 0

Setting each factor equal to zero gives us two possible values for y:

y + 12 = 0, which leads to y = -12.
y - 5 = 0, which leads to y = 5.
So, there are two possible pairs of numbers that satisfy the given conditions:

x = 7 and y = -12
x = 7 and y = 5
Both pairs make the equations true, and the original problem has two valid solutions.

Explanation :  

When it is 12 pm, adding five hours makes it 5 pm.

Answer: 12 kids

Explanation :  

We have the following information:

10 kids had juice (J).
8 kids had cake (C).
6 kids had both juice and cake (J ∩ C).
To find out how many kids were at the party in total, we can use set theory and the principle of inclusion-exclusion:

Total kids = (Kids with juice) + (Kids with cake) - (Kids with both juice and cake)

Total kids = 10 (J) + 8 (C) - 6 (J ∩ C)

Now, let's adjust the equation based on the information you provided:

Since 6 kids had both juice and cake, there are 8 - 6 = 2 kids who had cake (C) but not juice.

So, the adjusted equation becomes:

Total kids = 10 (J) + 2 (C, without J) - 6 (J ∩ C)

Now, plug in the values:

Total kids = 10 + 2 - 6
Total kids = 12

Therefore, there were 12 kids in total at the party. This method uses set theory and the principle of inclusion-exclusion to find the total number of kids, considering both the ones with juice and cake and those with only cake.

Answer: 600

Explanation :  

We have the following information:

5% of the men wear one earring.
The other 95% are divided into two groups: half wear two earrings, and half wear none.
Let's calculate it step by step:

First, we find out how many men wear one earring:

5% of 600 = (5/100) * 600 = 30 men.

Now, we know that 95% of the men belong to the other two groups (those who either wear two earrings or none). To find out the total number in this group, we calculate:

95% of 600 = (95/100) * 600 = 570 men.

As you correctly pointed out, half of the 95% (570) wear two earrings, and the other half wear none. Since they all belong to the same 95%, you can consider it as if they all wear one earring.

Now, to find the total number of earrings being worn in the club, we add the number of earrings worn by each group:

Total earrings = Earrings worn by men with one earring + Earrings worn by men with two earrings or none.

Total earrings = 30 (one earring) + 570 (effectively one earring) = 600 earrings.

    Answer:Bill is 40, and Bill Jr. is 12.

    Explanation :  

    Eight years ago, Bill Jr. was x years therefore Bill was 8x

    Today

    (x+8) +(8x+8) = 52

    x+8 + 8x+8 = 52

    9x + 16 = 52

    9x = 52–16

    9x = 36

    Therefore x = 4

    Bill Jr. is

    x + 8 = 12

    4 + 8 = 12 years old

    Bill is

    8x + 8 = 32 + 8

    32 + 8 = 40 years old

    Therefore 12 + 40 = 52 years.

Explanation :  

Shape the 3 matches into a roman numeral four.

Answer:100 

Explanation :  

We have two trains:

1. Train 1 is traveling at 60 mph.
2. Train 2 is traveling at 40 mph.

One hour before they pass each other, they have been traveling towards each other for one hour. In that time, they will have covered a distance equal to their combined speeds.

So, the total distance covered by both trains together in one hour is:

Distance = Speed x Time

For Train 1: Distance covered by Train 1 = 60 mph x 1 hour = 60 miles

For Train 2: Distance covered by Train 2 = 40 mph x 1 hour = 40 miles

Combined distance covered by both trains = 60 miles (Train 1) + 40 miles (Train 2) = 100 miles.

Hence, one hour before they pass each other, the two trains are 100 miles apart. This illustrates how understanding the concepts of speed, time, and distance allows us to analyze and solve real-world scenarios involving moving objects.

Explanation :  

One way to solve this math  is to use even numbers: The elder sister will be twice as old as her younger sister in three year's  time. This immediately rules out the elder sister currently being 8, 11, and 14, so she must be 17, and the younger sister 7.

   Answer: 6,457

   Explanation :  

Let's say you have a number represented as "ABCD," where A, B, C, and D are digits. In the given series:

1. The last digit (D) is moved to the front.
2. The other digits (A, B, and C) are shifted one place to the left.

So, mathematically, the transformation can be represented as:

Original Number: ABCD Transformed Number: DABC

In this way, the last digit is moved to the front to create the next number in the series. For example:

1. For 7,645:

   • Last digit D = 5
   • The remaining digits ABC = 7,64
   • So, the transformed number is 5,7645.

2. For 4,576:

   • Last digit D = 6
   • The remaining digits ABC = 4,57
   • So, the transformed number is 6,457.

You can use this mathematical pattern to generate the next number in the series by applying the same rule: moving the last digit to the front and shifting the other digits one place to the left.

Explanation :  

Four brothers and three sisters.

Explanation :  

Let the two whole, positive numbers be represented by $a$ and $b$. The given conditions are:

1. When multiplied, the result is a one-digit number: $a\times b=\text{Single Digit}$

2. When added, the result is a two-digit number: $a+b=\text{Two Digit Number}$

Now, let's consider the specific case of $a=1$ and $b=9$.

1. Multiplication: $1\times 9=9$

   The result is a single-digit number, which satisfies the first condition.

2. Addition: $1+9=10$

   The result is a two-digit number, fulfilling the second condition.

Thus, the numbers $a=1$ and  $b=9$ meet both criteria. This demonstrates the mathematical solution to the given puzzle, where the multiplication of 1 and 9 yields a single-digit result, and the addition of 1 and 9 produces a two-digit sum.

Explanation :  

Let's represent the arrangement using variables:

$a=888(\text{Three eights})$

$b=88(\text{Two eights})$

$c=8+8+8(\text{Three individual eights})$

Now, the given equation is:

$a+b+c=1000$

Substituting the values of $a$, $b$, and $c$:

$888+88+8+8+8=1000$

This shows the breakdown of the arrangement into its individual components. The combination of three eights, two eights, and three individual eights results in a sum of one thousand.

Thus, the mathematical explanation demonstrates how each part of the arrangement contributes to the total sum, fulfilling the condition $888+88+8+8+8=1000$.

Explanation :  

The total of the mother's age and the daughter's age is 50.
5 years ago, the mother was 7 times the age of the daughter.
Therefore,
If the total of their ages is 50 now, then 5 years ago the total of their ages was 40 (50 years minus 5 years off the mother's age and 5 years off the daughters age, or 50 (5+5)).
That last point is what will help us solve this problem. Let's state the last point as an equation. For the purposes of crafting some nice tidy equations,

M=Mother's age 5 years ago
D=Daughter' age 5 years ago
The first equation is pretty simple:
M+D=40
We also have to write the part about the mother being 7 times older than the daughter into an equation:
M=7D
Now that we have two equations, we can use them together to solve the question about how old the mother and daughter were 5 years ago. 
First, we can get rid of M to make the equation easier to solve.
Since M=7D, substitute the M in the first equation with 7D in the second equation:
M+D=40  > 7D+D=40
Now solve for D.
7D+D=40
8D=40
8D/8=40/8
D=5
So 5 years ago, the daughter was 5 years old. 
Now extrapolate:
5 years ago the daughter was 5, so now she's 10.
5 years ago, the mother was 7 times the age of her daughter (7D=M, or 7(5)=M), 
So, 5 years ago the mother was 35, and now she's 40 (7D+5=Mother's current age, or 7(5)+5=Mother's current age)
And if you need more proof,
40+10=50