FOR KIDS WHO LIKE TO SOLVE PROBLEMS

#5
CONTENT It is good to have an end
to journey toward; but it
Tickets for Mars 2 is the journey that matters,
in the end.
~ Ursula K. Le Guin

Count Two Ways to Save the Day 6

There is no such thing as a “math person.”

Bacteria Brouhaha 8 However, a child’s early learning experiences
directly impact his perception of his mathematics
ability. When children are taught mathematics
THIS MAG
AZINE
through fun and engaging problems, they not only BELONGS
TO:
MChallenge 10 build fluency in and an intuitive understanding
of mathematics, but they also develop critical
problem-solving skills.
Clever Climber Conundrum 12
MPower! was created as a resource for children Not all those who
who want to flex their mathematical muscles wander are lost.
and show off their problem-solving ingenuity.
~ J.R.R. Tolkien
We hope you enjoy the stories and problems
MTwist 14
inside.

KEY
Observing the Snail 16
GRADES 1 & UP
GRADES 3 & UP
GRADES 5 & UP
Algebra in Pictures 18 MASTER SOLVER

The universe is a big

Answers 21 place, perhaps the biggest.
~ Kurt Vonnegut
Connect with us at: MPower@russianschool.com
 Ladies and gentlemen, children of all ages! Welcome aboard. Our destination today Deimos
is Mars, the mysterious red planet, our close neighbor in the solar system. Ah, but we
Phobos
are not headed to the barren, dry desert of today, but to an alternate Mars of millennia
ago, bursting with vegetation and teeming with life. Expect to run into creatures of all
shapes and sizes! For this is not an ordinary vessel — it is a portal to a parallel universe!

Buckle up for the ride! Our first stop is Valles Marineris, the largest canyon in our solar
system. Have you ever seen the Grand Canyon in Arizona? It is super small compared
to Valles Marineris! You can easily fit dozens of Grand Canyons in Valles Marineris, which
is 7 km deep in places. Doesn't it look like just a giant crack running along the surface
of Mars? It is! The surface fractured when it was cooling off about 4 billion years ago.
Just imagine the planet's crust splitting apart and red-hot lava rushing out through
the fissures that were thousands of kilometers long! But no worries, we are not going
to visit those dangerous times but will instead visit Valles Marineris when it was home
to a friendly tribe of three-armed Martians.

Look, here are 15 Martians sitting in a row holding hands! Each Martian
holds one hand of the neighbor on each side. How many hands remain
free, not holding another hand?

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 Our next stop is Olympus Mons, the tallest mountain not only on Mars but in the solar Our last stop is Vastitas Borealis, the largest lowlands of Mars. We are heading into
system! It is 2.5 times as high as Mount Everest, which is the highest mountain on Earth. the most ancient times, billions of years ago! Did you bring your swimsuits? A huge ocean
Olympus Mons would cover the entire state of Arizona if we moved it to Earth, covered a third of the planet back then. But if a member of a mission to Mars today needs
but no worries, Arizona, we’ll keep it here on Mars! ice for a smoothie, he'll be able to find it here. The red planet has lots of frozen water now.

Time to put on your protective gear! Today is a warm summer day, so it is a nice
Two friends, Zack and Zelda, are gathering pretty pebbles on the beach.
and comfortable 70°F on the surface. But be ready to fight the almost cosmic cold on top
The friends collect the same numbers of pebbles. If Zack gives three
of Olympus Mons. It will be –30°F up there, which is very warm compared to outer space,
of his pebbles to Zelda, how many more pebbles does Zelda now have
but the atmosphere is so thin it will feel like we’re in outer space!
compared to Zack?

Here are three children picking berries in the foothills of Olympus Mons.
Each child has 20 berries. The first child eats some of his berries and the second
child eats as many as the first child has left. Then the third child eats as many
as the first and the second eat combined. How many berries remain?
Olymp
us Mon
s (21,229
m)

Some Mars rovers move incredibly slowly – Phobos – one of the moons of Mars – From Mars, Earth looks like
about 30 meters per hour. Rovers need is doomed! It is getting closer to the Martian a bright green star that And you can see a dim
this time to calculate every move! surface, and in just 100 million years, Phobos is visible in the morning yellow star next to Earth,
will be ripped apart by the tidal forces of Mars. and evening skies. which is our Moon. What year will humans go to Mars?
You are as fast as I am! Part of the breakup material will form a ring NASA hopes to launch astronauts
around Mars, and part will plunge onto to Mars by the late 2030s, while
the planet’s surface. SpaceX plans a manned mission
to Mars even earlier, by 2030.

4
8 5
 But wait, we never
used the 7 and 5 from
the problem! This RSM
gang is always trying
to trick us!

This is sooo confusing,

I get a headache when
I start thinking about it.
I’d better draw a picture.
Hmmmm, I think that
humpties look like this…

Oh, wow, listen! You must have

the same total! It is the number
of friendships, the number
of golden strings between
Hold on, I have an idea of how we can make the picture more
humpties and dumpties!
organized! Look, we can put humpties on the left and dumpties
Oh, I get it! So, 9H = 10D
on the right! And we can show friendships with golden strings
and it is clear that H > D.
that connect humpties and dumpties. All right, all right, if you
don’t have any golden string, we can just draw lines. Okay, I started on the left. Nine lines
go from each humpty to the other side.
Let’s say we have H humpties.
We will never solve this So, 9 · H friendships.
problem if we draw all
the humpties and dumpties,
so let’s just use dots I don't agree! I’m a lefty, so
and letters! Remember how I started on the right! So, it
Ms. Clever told us that’s is the other way around for me!
what mathematicians do? Each dumpty has 10 lines that go
So, I started drawing dots, to the other side. Let’s say we
but I made a terrible mess! have D dumpties.
So, 10 · D friendships.
6 7
 Never fear, my young colleague! We shall be wiser
this time. Two things we can do: first, we will use
yogurt bacteria as our subject, and second, we will
set up a warning alarm to let us know when to stop
the experiment. That way, we should prevent
a mishap... but if we don’t, at least we will end up
B-b-b-but, Professor!... Are you sure this is a good idea?
with a roomful of delicious yogurt!
Don’t you remember what happened the last time we tried to grow
bacteria in a jar? It overflowed, and we were washing green slime
out of our equipment for weeks! I still have nightmares about it!
Let’s see... we know that the bacteria population
doubles every second and that it takes one hour to fill
up the jar. How about we set the alarm to go off when
___ 1
the jar is just over ​​ 10   ​​full? That should give us plenty
of time to safely stop the experiment.
Let’s put a single bacterium in the jar at exactly 10 AM.

In one second, we will have two bacteria... in two seconds, we will have four...
in three seconds, we will have eight…

in 1 sec. in 2 sec. in 3 sec.

In one hour, the jar will be full! Aack! Professor, Act quickly,
the alarm is buzzing! my trusted assistant!

When did the warning alarm sound? Do you think the professor and her
colleague were able to stop the experiment before the bacteria escaped the jar?
Or did they end up with yogurt on their faces?
8 9
 4) What is the mass of one flying saucer? 5) Complete the Magic Square. Write
the numbers 4, 6, 9, 10, 11, 12 in the empty cells
so that the sum of the numbers in each row,
each column, and each diagonal is equal.
1) A pirate has two treasure chests. The first chest 3) A salt solution has mass 8 kg 462 g
contains 100 gold coins and the second chest and water is 99% of the solution.
is empty. Each day, the pirate puts one gold coin How much water needs to be removed
in the first chest and two gold coins in the second by steaming so that the resulting
chest. In how many days will the number of coins solution contains 98% water?
in the chests be equal?

6) A rectangle and a square have equal

perimeters. Does the rectangle or the square
2) Paul throws two darts at the round target shown.
have larger area?
Which option shows a total number of points that
is impossible for Paul to score with two throws?
(A) 6 (B) 8 (C) 9 (D) 10 (E) 12

10 11
 A mountain climber sets out on a climb at 8:00 AM and reaches the top
of the mountain at 8:00 PM. After spending the night at the summit, he
begins his descent at 8:00 AM. He uses the same route to descend as he
used to ascend the previous day. The climber reaches his original starting
point at the bottom of the mountain by 8:00 PM. He says that he passed
every point along the route twice — once on the first day when ascending
and once on the second day when descending. However, the mountain
climber claims that he was never in the same spot at the same time of day.
Is this possible, or is he wrong?

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 For your mathematical
exploration, you will need:

During a NASA exploration, a rover on Mars received the directions below. Study the first Cut this cake along the grid lines into four
expedition. Then follow the directions and draw the rover’s path for the second expedition. pieces. Make the four pieces the same size
and the same shape, and of course each needs
EXPEDITION 1 EXPEDITION 2
one chocolate and one cherry on top!
3↑ 3← 3↑ 7→ 7↑ 1→ 1↓ 1→ 1↓ 1→ 1↓ 1→ 1↓ 1→ 1↓ 1→ 1↓ 1→ 1↓ 1→ 7↑
3↓ 3← 3↓ 3→ 7↓ 7↑ 4→ 4↓ 4← 4→ 3↓ 6→ 3↑ 4← 4↑ 7→ 7↓
4↑ 4→ 3↑ 4← 3↓ 4→ 4↓

Which ropes will make a knot if you pull the two ends? Choose all correct answers.
Start Start

A B C

Study your dice. Then use the picture to find

the total number of dots that you cannot see.
D E F

14 15
 Is it possible that the snail moves 3 meters in 48 hours?

Let’s see… The snail moves 1 meter every 24 hours. She moves 1 meter during the first 24-hour period,
Summer days are hot and slow. Sometimes, all that one wants
and another meter during the second 24-hour period. That’s just 2 meters in total, so 48 hours
to do is lie in the shade and lazily watch a snail move at its own
Study the round ripples is not enough time for her to travel 3 meters.
pace. The source of the wisdom quoted on the right may not be
in the water. Otherwise, tossing
a particularly serious individual, but we treat his advice with
stones is but an empty pastime.
the utmost attention. To us, this brief statement embodies What if we give the snail just a little bit more time, say 48 hours and 1 minute?
the following: Observe the world around you, be an explorer Is this enough time for her to move 3 meters?
~ Kozma Prutkov, a pen name
and a problem solver. Allow the world to intrigue you, to motivate
used by four Russian poets
you, and to teach you how to observe and to question.
Surprisingly, this changes everything! Indeed, the speed of the snail is not necessarily constant.
For example, the snail could have a burst of energy from 8:00 AM to 8:01 AM and travel 1 entire meter.
Without further delay, let’s follow the snail! Consider a classic problem.
She then rests for almost 24 hours, until 8:00 AM the following day. Being a well-organized snail,
she does the same every 24 hours. So if we start timing at 8:00 AM, the snail moves 3 meters
A snail needs to go to the top of a tree that is 10 meters tall. Each day, she moves in 48 hours and 1 minute.
4 meters up. Each night, she moves 3 meters down the tree trunk. When will the snail
reach her destination if she starts her journey on Monday morning?
If 1 minute changes everything, then what about a distance that is longer than 3 meters?
Can the snail move farther than 3 meters in 48 hours and 1 minute?
It’s tempting to say that the snail moves up 1 meter every 24 hours, and so will travel 10 meters in 10 days
(or ten 24-hour periods). But actually, our snail will reach the tree top on Sunday evening. She moves
We know that the snail moves forward 3 meters in 72 hours. She cannot move backward,
up 1 meter every 24 hours beginning at any point in time, so for example, from 2:56 PM on one day
and in 48 hours and 1 minute she will move no more than 3 meters. Adding 1 minute did not help,
until 2:56 PM the next. Surprisingly, if we give our snail less than 24 hours, she may move farther!
but is it possible that allowing the snail to move backward will help? Suppose the snail is able to move
both forward and backward along a given straight line, and all other conditions of the problem remain
How interesting! Are there other ways to make our snail a race champion? What if we forbid her from
the same. How far can the snail move in 48 hours and 1 minute?
moving down the tree? Will she go farther? Let’s observe this disciplined snail. Let’s say she moves
along an infinite straight line and travels 1 meter every 24 hours. That’s all we know. Explore
the possibilities! A snail moves with variable (not constant) speed along a straight line. She can move
both forward and backward. However, she moves forward 1 meter in a 24-hour period
that begins at any point in time. How far can the snail move in 48 hours and 1 minute?
Observing th
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16 17
 Isn't math-in-art elegant? We examined artwork of simple shapes that are made of unit squares
ALGEBRA IN PICTURES and obtained a famous formula for the sum of natural numbers. Without the art, we’d have
needed complicated methods such as mathematical induction to write a proof.
Now we are ready for more complex, and hence more exciting, problems. It’s time to draw more!
Legend has it that the composer Salieri tried to “check harmony using algebra.” Could we do the opposite
and check algebra using art? What if we depict formulas with pictures? But we don’t mean drawing
random doodles, as much fun as that would be. Today we’ll try something incredible – let’s see if we
PROBLEM 1 PROBLEM 2
can replace ordinary calculations and derivations with pictures! We will focus on the different formulas
Draw your own picture to explain Picture 3 shows L-shaped corners. Can you spot
of sums, which are usually proved by sophisticated methods.
the identity. the pattern and find the number of squares
Let’s start with a simple example and before we know it, we’ll get to complicated and unexpected ​ ​
1 + 2 + … + (n − 1) + n + (n − 1) + … + 2 + 1 = ​n​​ 2​ in the next corner? How many squares
are in the 10th corner? How many squares
formulas, which are quite unbelievable and are difficult to prove even using algebra. By the way, all our
are in the nth corner?
visual proofs will be really short, they will pretty much boil down to the phrase "look carefully
at the picture..."

Picture 1 shows a geometric shape. Use two of these shapes

to form a rectangle. What are the dimensions of this rectangle? Picture 3
Do the same with the shape shown in picture 2 to form
a bigger rectangle. What are its dimensions?

Picture 1 Picture 2 Picture 4

Look at picture 4. You can easily see that each

Let's use this idea to get the formula for the sum of natural
corner has two more squares than the previous
numbers from 1 to n.
corner. So, each new corner has an odd number
of squares that is two more than the previous
How can we represent 1 + 2 + 3 + ... + n as a picture? Again,
corner. And so the nth corner has 2n – 1 squares.
with a staircase! When two such staircases are put together,
they form a rectangle with dimensions (n + 1) × n.
PROBLEM 3
(n + 1) ⨯ n
Therefore, each staircase has ​​  __​ unit squares, which
 ​
 
2
Look carefully at picture 5.
shows the sum of all the natural numbers from 1 to n.
After studying it, can you tell
(n + 1) ⨯ n what is the sum of the first
1 + 2 + 3 + ... + n = ​​  __​ 
 ​
 
2 n odd numbers?
Picture 5
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 PROBLEM 4
The following formula is sometimes called n=4
the reversionary sum of the first n odd numbers. 1+3+5+7+5+3+1=3 ​  2​ ​  2​
​ ​​ ​ + 4
​ ​​ ​ Tickets for Mars (p. 2) MChallenge (p. 10)
On one hand, we could decide to develop this 17 hands remain free. 1) 100 days 2) (C) 5) 5 10 9
formula from the previous problem. But on the other 20 berries remain. 3) 4 kg 231 g of water needs to be removed.
12 8 4
hand, we could use picture 6 to arrange two squares Zelda has 6 more pebbles than Zack. 4) The mass of one flying saucer is 675 tons. 7 6 11
and develop an independent proof-in-art. 6) H int: Let a be the side length of the square.
Count Two Ways to Save the Day (p. 6) Then the sides of the rectangle can be expressed
​
1 + 3 + … + (2n − 1) + … + 3 + 1 = ​(n − 1)​​  ​ ​  ​
2
​ ​​ 2​
​+n Warm-up: It is not possible. as a + x and a - x for some x.
Picture 6 Bonus problem hint: The total number of hands must The square has a larger area.
be even because this number is equal to twice
the number of connections. Clever Climber Conundrum (p. 12)
PROBLEM
PROBLEM5 5 On the comment from the flying creature: Indeed we did Hint: Try to schematically plot both his ascent and descent
not use the numbers 7 and 5! These numbers would help on the same time-distance graph.
How many more squares are in the outer frame than
us find lower boundaries for the numbers of humpties The mountain climber is wrong.
in the inner frame?
and dumpties, but they do not help us solve this problem.
Observing the Snail (p. 16)
Bacteria Brouhaha (p. 8) Amazingly, a snail can travel as far as it wants in 48 hr 1 min!
There are 8 more squares, and there is no need
11:00:00 am jar is full Suppose this distance is 1 km. Then the snail can rush
to explain this in words, the picture is enough.
10:59:59 am jar is _ ​ ​21 ​​ full 998 m to the right in the first minute, and then 997 m
This is not a hard problem on its own, but it will help us
10:59:58 am jar is __ ​ ​41  ​​ full to the left in the last 59 min of the hour. So, in any 1-hour
with the next problem. 1
Picture 7 ​  8  ​​ full
10:59:57 am jar is ​__ period, the snail moves 1 m to the right. Thus, in 2 hours,
10:59:56 am jar is __
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​ ​ 1  ​​ full the snail is 2 m right of its starting point. In the next minute,
1 it rushes another 998 m to cover a total of 1 km. This strategy
PROBLEM 6 So, the jar will be ​​ __     ​​ full between 10 hr 59 min 56 sec
10 works for any distance!
and 10 hr 59 min 57 sec.
Study picture 8 to explain the identities and continue the sequence. The professor and her assistant have only about
Algebra in Pictures (p. 18)
3 seconds to stop the experiment, and it may
1×1=1 Problem 1: Problem 3:
3×3=1+8
or may not be long enough!
​
1 + 3 + 5 + … + (2n − 1) = ​n​​ 2​
5×5=1+8+2×8
MTwist (p. 14)
7×7=1+8+2×8+3×8
1) 3)
9×9=1+8+2×8+3×8+4×8
11 × 11 = ? Problem 6:
13 × 13 = ? Picture 8 11 × 11 = 1 + 8 × 1 + 8 × 2 + 8 × 3 + 8 × 4 + 8 × 5,
15 × 15 = ? 2) Hint: What is the sum of the dots on one die? 13 × 13 = 1 + 8 × 1 + 8 × 2 + 8 × 3 + 8 × 4 + 8 × 5 + 8 × 6,
25 dots are not visible on the dice shown. 15 × 15 = 1 + 8 × 1 + 8 × 2 + 8 × 3 + 8 × 4 + 8 × 5 + 8 × 6 + 8 × 7,
20 4) Ropes A, B, and F will make a knot. (2n + 1) × (2n + 1) = 1 + 8 × 1 + 8 × 2 + ... + 8 × n. 21
 Team RSM at Harvard – MIT
Mathematics Tournament (HMMT) 2021

Adam Ge Selena Ge Ali El Moselhy Aryan Raj Robert Rogers

Sebastian Prasanna (Not Pictured)

1st in the Guts Round

7th out of 150 overall

Congratulations to Team RSM and coaches for your performance

at the most exclusive high school math competition in the world.
We’re so proud of you!