## Previous Challenges

### Here is a timeline of previous challenges and their solutions.

16/12/2019

#### If $6\cdot4=26$, $6\cdot5=33$, $6\cdot6=40$, and $6\cdot7=46$, what does $4\cdot5\cdot6=$?

143

Why? Well if we look at this logically, there is a clear base conversion here. When we have base $N$, we know that the way to convert it back into decimal ($\text{Base}_{10}$). We multiply each digit by the base raised to the degree of one less than that digit's place.

Example: $2012_{3}\to\text{base}_{10}$
$\to 2(3)^{3}+0(3)^{2}+1(3)^{1}+2(3)^{0}$
$\to 2(27)+0(9)+1(3)+2(1)$
$\to 54+0+3+2$
$=59$

So, where in decimal, the answers would be $24,30,36,42$ respectively, we get the sequence $26,33,40,46$. So we work through what we know.

$24_{10}\to 26_{N}$
Well, let's write it out like we did above so that $2(N)^{1}+6(N)^{0}=24$
$\to 2N+6=24$
$\to 2N=18$
$\therefore N=9$.

It is all in base 9. We can use a base-converter like the one below to do this, or we can use our knowledge of base-conversion. For now though, let's just use the tool below.

 base-10: base-9:

09/12/2019

#### What is the greatest prime factor of $4^{19}-2^{34}$?

$4^{19}-2^{34}$
$\to 4^{19}-4^{17}$
$\to 4^{17}\left(4^{2}-1\right)$
$\to 4^{17}\left(16-1\right)$
$\to 4^{17}\left(15\right)\to 4^{17}\left(5\times3\right)$
$\to 2^{34}\times 3\times 5$

So the higest prime factor is $5$.

02/12/2019

888+88+8+8+8.

Easy.

## November 2019

25/11/2019

#### If $5+3=16$ and $7+3=40$, then $9+3$ must equal what?

$72$.
Why?

$5+3\to5^{2}-3^{2}=25-9=16$.
$7+3\to7^{2}-3^{2}=49-9=40$.

$\therefore \ 9+3\to9^{2}-3^{2}=81-9=72$
Q . E . D

18/11/2019

#### The graph $f(x)=ax^{2}+bx+c$ cuts the $y$-axis at the point $(0, 2)$ and the $x$-axis at the points $(1, 0)$ and $(2, 0)$. Find the value of $a, b, c$.

Well, $c$ is the $y$-intercept, which is given as $2$. We know that when $x=1$ and $x=2, y=0$. Therefore we have two equations with two unknowns, $a$ & $b$. We can easily solve this then using Junior Certificate simultaneous equations.

$a+b+2=0 \to b=-a-2$
$4a+2b+2=0 \to 4a+2(-a-2)+2=0$
$\to 4a-2a-4+2=0$
$\to 2a-2=0 \to 2a=-2 \to a=-1$
$b=-1-2 \to b=-3$

So... $a=-1, b=-3, c=2$.

11/11/2019

#### What is the second most irrational number?

The most irrational number out there is $\phi$. This is followed by the second most irrational number, $\sqrt{2}$.

04/11/2019

## October 2019

28/10/2019

#### Find $y$ when $4y-2=2$.

$4y-2=2 \rightarrow 4y=4 \rightarrow y=1$.

21/10/2019

#### What are the odds of picking a picture card or an ace out of a standard deck of playing cards?

There are three picture cards and an ace per suit. That's four.
There are four suits.
That's four times four - sixteen.
There are fifty-two cards in a standard deck.
Sixteen out of fifty-two is the same as $\frac{4}{13}$.

14/10/2019

#### Find $p$ when $3(4p-7)=7p+9$.

$12p-21=7p+9 \rightarrow 5p=30 \rightarrow p=6$..

07/10/2019

#### The perimeter of a rectangle is 18cm. If it has a width $2x$cm and a length $(x+3)$cm, find the value of $x$.

$2(x+3)+2(2x)=18 \rightarrow 4x+2x+6=18$
$4x+2x+6=18 \rightarrow 6x=12 \rightarrow x=2$.

## September 2019

30/09/2019

#### What famous mathematician worked on calculus and came up with three famous laws of motion?

Sir Isaac Newton.

23/09/2019

6886

16/09/2019

#### If a passcode has to be ten digits long, only numbers, and no number can only be repeated twice, how many possible passcodes could there be?

$20\choose10$ $=184756$

09/09/2019

3

02/09/2019

#### Solve for $x=\sqrt{19-2x}+2$

$x=5$ is the only solution.

## August 2019

26/08/2019

#### Solve for $x-11+\frac{24}{x}=0$

$x=3$ or $x=8$

19/08/2019

#### Solve for $x$ when $\frac{1}{27}\cdot3^{59}\cdot\frac{1}{243}\cdot27^{x}=\frac{1}{3}\cdot3^{x}$

$x=-26$

12/08/2019

#### How many numbers less than 1,000,000 have digits that total 3 (e.g, 12, 300, 2001)

$8\choose3$ or 56.

05/08/2019

#### At a certain school, the ratio of first year students to third year students is 4 to 7. The ratio of transition year students to first year students is 3 to 4. The ratio of second year students to third year students is 6 to 5. What is the ratio of transition year students to second year students?

5 to 14. Work it out if you don't believe us. To see the workings, check the M4TH5 Forums.

## July 2019

29/07/2019

#### Solve for $3x -5 = 16$.

$3x = 16+5 =21$.
$x=\frac{21}{3}$.
$x=7$.

22/07/2019

#### What do 8 and -9 have in common?

They both satisfy the equation $x^{2} +x =72$. Can you explain why there are infinite relationships like this?

15/07/2019

#### Which mathematician's name contains "bob" and arm"?

The infamous unabomber.

08/07/2019

#### Divide $x^{2}+5x+6$ by $x+3$

$x+2$.

$x^{2} \div x = x, \ x\left(x+3\right) = x^{2}+3x$.

$x^{2} +5x -\left(x^{2}+3x\right)=2x$.

$\frac{x^{2}+2x}{x}=x+2$.

01/07/2019

#### The sum of a number and half of the number is 12. Find the number.

8.

$8+\frac{1}{2}8=12$.

## June 2019

24/06/2019

#### Express $\frac{3}{5}-\frac{2x-1}{10}+\frac{3x-2}{4}$ as a single fraction.

$\frac{11x+4}{20}$.

17/06/2019

#### A contractor estimated that he could do a certain job in 1 year with 280 men. If he were asked to do the work in 10 months, how many more men would he need to employ?

56... we'll let you figure this one out.

10/06/2019

#### Why is 1 not a prime number?

Ultimately, a prime number can be defined as a natural number with exactly four factors.

1) Positive itself
2) Negative itself
3) Positive one
4) Negative One

The number 1 only has two factors. Not four. It shares the pairs of positive factors and also the negative factors. This is why one is not a prime number.

03/06/2019

#### Solve the inequality:4-3x > -5, $x \in \mathbb{Z}$

$4-3x>-5$.

$\ -3x>-5-4$ ... take 4 from each side.

$\ -3x>-9$.

$\ \ \ 3x<9$ ... multiply both sides by -1 and reverse the inequality sign.

$\ \ \ x<3$ ... divide both sides by 3.

$\therefore x = 2, 1, 0, -1, -2, \ldots$.

## May 2019

27/05/2019

#### How many three-litre bottles of water can you fill with seven two and a half beakers of six-hundred litres, each a quarter filled?

$7 \times 2.5 = 17.5$.
$17.5 \times 600 = 10,500$.
$10,500 \times 0.25 = 2,625$.
$\frac{2,625}{3} = 875$.

20/05/2019

#### What is the most complicated way of solving 16 + 16?

Well, this can't really be answered. There is always a more complicated way.

We would approach this (while trying to save time) by splitting this down to the most simple units and adding them up individually.

16 + 16 = 8 + 8 + 8 + 8.

8 + 8 + 8 + 8 = 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4.

4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2.

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.

Now if you count the number of one's, you get the answer. 32.

So 16 + 16 = 32.

Obviously there are many many more ways of proving this that are more complicated. But this is our simple yet slightly (very slightly) complicated approach to the question.

13/05/2019

#### What is 5 + 7?

Yes. A simple addition question. Not all of these need to be difficult you know. Some are really simple. Like this one.

5 + 7 = 12.

06/05/2019

#### Is it possible to square the circle?

No.

The circle has an area of $\pi \ r^{2}$.
A square has a area of $L \times W$. This means that the length of one of the squares sides equals a multiple of $\pi$. This obviously is not possible as $\pi$ is an irrational number. So, no, you cannot square the circle.

## April 2019

29/04/2019

#### WARNING - THIS CHALLENGE INCLUDES DIVIDING BY ZERO If, $\frac{x}{0} = x$ what does $\frac{0}{x}$ equal?

4.

Why? Well...
if $\frac{x}{0}=x$, then $x=0x$ (multiply each side by zero).
$0x=0$, so, $x=0$.

Therefore, $\frac{0}{x}= \frac{0}{0}$.
$\frac{0}{0}=4$ because when you multiply both sides by zero, you still get zero.
$0=0(4)$

Likewise, the answer is also 5, and 3, and 6, and any natural number you can think of.

Actually, $-1(0) = 0$, so it's not just natural numbers. It's intergers. Except, $2.5(0) = 0$, so it's any rational number.

Hold on, $\pi(0) = 0$, so it's any real number, and that's the answer.

But hold on,  $\sqrt{-1}(0) = 0$ also. So it's any complex number.

In fact, it is undefined. There are too many different numbers it can be that we do not consider it a number. Some may consider it as one, but that doesn't make sense algebraically. We just don't do it. It brakes math as we know it.

22/04/2019

#### Using three 2s and any math symbols and/or operations you choose, can you create 25?

$\sqrt{.2^{-2}}^2$. That's one way we found. We know that looks confusing though.

Here it is in a slightly less ... confusing way.

15/04/2019

#### Solve these systems for K. P + Q=6 | KP + Q = 18 | P + KQ = 30

KP + Q + P + KQ = 30 + 18 ... adding the final two systems.
K(P + Q) + P + Q = 48.

And since P + Q = 6, we get K(6) + 6 = 48.
K(6) = 42.
K = $\frac{42}{6}$.
K = 7.

08/04/2019

01/04/2019

## March 2019

25/03/2019

#### A positive whole number is divisible by 3 and also by 5. When the number is divided by 7, the remainder is 5. What is the smallest number that could work?

If it is divisible by 3 and also 5, then it must divisible by 3x5 = 15.

Since:
15 = 7 x 2 + 1
1 is the remainder. We want a remainder 5. So let's multiply the equation by 5.

15 x 5 = 5(7 x 2 + 1)
75 = 35 x 2 + 5

18/03/2019

#### If a + b + c = x and a x b x c = x, and each of a, b and c is a positive integer less than 10, find x.

a + b + c = x = abc, so
a + b + c = abc.
c = abc - a - b.
Note: kc = 2kc - kc, where k is a constant.

Since kx = 2kx - kx
kc = 2kc - kc

The coefficent infront of c in [c = abc - a - b] is 1, so in total k equals 1, but let's prove this.

So abc = 2kc, and -a-b = -kc.
So ab = 2k and -a-c =-kc.

As above:
c = abc - a - b.
c = 2kc - kc.
Divide by c,
1 = 2k - k
1 = k

Put back into other equations.
abc = 2kc
ab = 2k
ab = 2(1)

So either a or b = 2, and the other = 1, it doesn't matter in this case.

1 + 2 + c = 2c
3 + c = 2c
3 = 2c-c
3 = c
1 + 2 + 3 =
(1)(2)(3) --> True
1 + 2 + 3 = x = 6

So x = 6.

11/03/2019

04/03/2019

#### How many ways can you select twelve different seconds from your life if you'll die when you're exactly 84 years old, given you were born on Feb 1st on a leap year?

Well, first you have to figure out how many seconds you have been alive for.

Well there's 60 seconds in a minute.
60 x 60 seconds in an hour.
3600 seconds.
3600 x 24 seconds in a day.
86400 seconds.
86400 x 365 seconds in a normal year.
31536000 seconds.
31536000 + 86400 seconds in a leap year.
31622400 seconds.

Next, how many leap years have you been alive for.
84$\div$4 = 21 years.
[If 85 years old, you'd divide 85 by 4 and round down. Same for 86 and 87. Unless the result is even, you round down, not up.]

Next, how many normal years were you alive for?
84 - 21 = 63 normal years.

So the number of seconds you were alive for is: [31536000 x 63] + [31622400 x 21]
2650838400 seconds you were alive for.

The 'n Choose r' function would tell you the answer, but if you put that into a calculator, it'll be too big. Luckily, thanks to the internet, there are High Precision Calculators.

We used this one to calculate the answer below: https://keisan.casio.com/calculator.

2.513402616591275829041783946019934930247267348224179879211788934269136263836500637385643571163009606E+104

## February 2019

25/02/19

#### If there were sixteen seconds in a minute, ten minutes in an hour, five hours a day and two hundred and sixty-one days a year, with no leap years, how far would a light year be in kilometers?

16 x 10 x 5 x 261 = (208,800) number of seconds in a year.

The speed of light is 299,792,458 m/s

299,792,458x208,800 m/year
62,596,665,230,400 m/year
Divide by 1000 to make km/year

So a light year in this case is 62,596,665,230.4km

18/02/2019

#### Find the pattern [e.g. T(n)=3n+5] in 4, 15, 40, 85, 156, 259.

To find out to what degree (power) this sequence deals with, we find the common difference.

4 15 40 85 156 259
\ / \ / \ / \ / \ /
11 25 45 71 103
\ / \ / \ / \ /
14 20 26 32
\ / \ / \ /
6   6   6

The 3rd difference is the same, so the sequence is in the 3rd Degree. This means that the sequence is written in the form An3 + Bn2 + Cn + D. So now we can use simultanious equations to find A, B, C, D.

T(1) = 4 = A(1)3 + B(1)2 + C(1) + D
= A(1) + B(1) + C(1) + D
= A + B + C + D

T(2) = 15 = A(2)3 + B(2)2 + C(2) + D
= A(8) + B(4) + C(2) + D

T(3) = 40 = A(3)3 + B(3)2 + C(3) + D
= A(27) + B(9) + C(3) + D

From here, we can calculate, through observation that
A, B, C and D = 1. So we can conclude that the formula is:
n3 + n2 + n +1

11/02/2019

04/02/2019

#### How many different games can you win Connect Four in?

Out of the 4,531,985,219,092 different states of which a traditional 7x6 Connect Four game can have, there are 1,905,333,170,621 (just under two trillion) different ways of finnishing, from litterally dropping four down in a straight line, to filling the whole board and drawing. Out of this there is 1,904,587,136,600 ways to win.

It takes a long time to calculate, but computers can analyse the system to determine the figures you see above. If you have written a programme that can solve this question, Please feel free to share it with us at info@m4th5.ie.

## January 2019

28/01/2019

#### How can $1+2+3+4+5+6+... =- \frac{1}{12}$?

Well, logically it doesn't. That's upfront obvious. If you cut off at any point in this infinite series, the result will be exponential. However, using certain maths, this can be proven in a mathematical sense.

See here.

21/01/2019

#### Prove that phi (1.61803398874989...) is the most irrational number.

Well, the most irrational number is the one with the most simple continuous fraction. This is:

1+(1/(1+1/(1+1/(1+1/(1+1/(1+1...))))))

If we make this x, we can see that after the first fraction line, x is repeated.

Therefore we can write this as:
x = 1+(1/x)
Multiply accross by x and we get:
x2 = x + 1
bringing accross [x + 1] and we get:
x2 - x - 1 = 0

Using the quadratic formula, we can simply determine that $x =$$1+\sqrt{5}\over 2$ which is phi.
this is better shown here.

14/01/2019

#### A rectangle has a tolerance interval of 15.5 +/- 0.5 cm for the length and a tolerance of 20.3 +/- 0.5 cm for the width. Find the percentage error for the length of the rectangle. Give your answer to 2 decimal places.

The percentage error for length is the error devided by the true value, multiplied by one hundred.
If the error = 0.5cm
And the true value = 15.5 cm
Then:

(0.5/15.5)x100=3.23%

07/01/2019

#### The sum of four numbers is x. Suppose that each of the four numbers is now increased by 1. These four new numbers are added together and then the sum is tripled. What is the value, in terms of x, of the number thus formed?

a+b+c+d=x
3(a+1+b+1+c+1+d+1)=3(a+b+c+d+4)
=3(x+4)
=3x+12