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A small step before this topic
Let’s picture it Let’s start with three stops.
Home. Then the park. Then the café. Just a few things in a row.
One little change Point to the middle one.
The park is second. In Python, its position is 1, because counting starts at 0.
Now it has a name That’s a list.
A list keeps things in order. An index is just a position in that list.
No pressure. Just try it.Which place is at position 0?
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Let’s picture it Now visit each stop.
You don’t need three different instructions. “Visit the next place” works each time.
One little change One, then the next.
Finish home. Move to the park. Then do the same at the café.
Now it has a name That repeat is a loop.
One small action, repeated for each item. Three stops mean three visits.
No pressure. Just try it.Add one more stop. How many visits now?
1 / 4 A small step before this topic
Let’s picture it What if it rains?
Let’s choose the café on rainy days, and the park on sunny days.
One little change Today is sunny.
The rule stays the same. The weather changed, so we choose the other place.
Now it has a name That’s an if / else.
Raining?
Yes: café
No: park
A condition is a yes-or-no question. Its answer chooses which step happens next.
No pressure. Just try it.It’s raining. Where does our rule send us?
1 / 4 A small step before this topic
Let’s picture it Name a tiny job.
Our job is “add one.” Give it 3, and it gives back 4.
One little change Try a new input.
Give the same job 8. It gives back 9. We can reuse the instructions.
Now it has a name That job is a function.
Input goes in. An answer comes back. In Python, return sends the answer back.
No pressure. Just try it.What does add-one return for 5?
1 / 4 A small step before this topic
Let’s picture it Picture a pile of plates.
Blue goes down first. Pink goes on top.
One little change Take the top one.
Pink comes off first, even though blue arrived first.
Now it has a name That’s a stack.
Last in, first out. Paused function calls work this way too: finish the newest call, then return.
No pressure. Just try it.Add green on top. Which plate comes off next?
1 / 4 A small step before this topic
Let’s picture it Now picture a line of people.
Ana arrives first. Bo waits behind her.
One little change Ana goes first.
Bo moves to the front. New arrivals join the back.
Now it has a name That’s a queue.
First in, first out. Unlike plates, we serve the oldest waiting item first.
No pressure. Just try it.Who goes before Cy?
1 / 4 A small step before this topic
Let’s picture it Have we visited the park?
Keep a little checklist of places we’ve seen.
One little change Write park again.
We still only need one check beside park. We’re remembering whether, not how many times.
Now it has a name That’s a set.
A set keeps distinct items. “Is park in here?” becomes a quick membership check.
No pressure. Just try it.Add park twice. How many copies in a set?
1 / 4 A small step before this topic
Let’s picture it Let’s remember a little more.
The park is five minutes away. Write that beside its name.
One little change Look it up later.
Ask for “park.” The notebook gives you 5. No need to measure the walk again.
Now it has a name That’s a dictionary.
A key finds a value. Here, the place name is the key and the time is the value.
No pressure. Just try it.We now write park: 6. What does park return?
1 / 4 A small step before this topic
Let’s picture it Draw our little walk.
A dot for each place. A line where we can walk directly.
One little change Two short walks can join.
Home to park, then park to café. We can reach the café without a direct road.
Now it has a name Dots and links have names.
A dot is a node. A link is an edge. Together, they make a graph.
No pressure. Just try it.Must two places share a direct road to be reachable?
1 / 4 A small step before this topic
Let’s picture it Where did we put the book?
Try “second shelf, first box.” We need two positions, not just one.
One little change Pick a row, then a column.
Python calls the second row 1. The first column is 0. So the book is at [1][0].
Now it has a name That’s a two-dimensional table.
Two labels find one cell. A third label could tell us which cupboard contains the table.
No pressure. Just try it.Two cupboards have the same shelves. What extra label helps?
1 / 4 A small step before this topic
Let’s picture it Same door. One small difference.
You have a key. Your friend doesn’t. The door is the same, but your next move isn’t.
One little change Remember the key.
If we only write down “at the door,” we lose something important.
Now it has a name That little note is a state.
A state remembers what we need to continue. Here: where we are, and whether we have the key.
No pressure. Just try it.Is “at the door” enough to decide if we can enter?
1 / 4 A small step before this topic
Let’s picture it How many little jobs?
Visit three places. Say hello once at each. That’s three hellos.
One little change Now try every pair.
Three people each greet all three people, including themselves for this toy count. That’s nine greetings.
Now it has a name Count how the work grows.
One pass grows with n. Trying every pair grows with n × n. Code length doesn’t tell us that.
No pressure. Just try it.With 4 items, how many ordered pairs, including self-pairs?
1 / 4 A small step before this topic
Let’s picture it Find 8 in a sorted row.
Start with 2, 4, 6, 8, 10. Look at the middle: 6.
One little change Keep the side that can work.
8 is bigger than 6. Everything on the left is too small, so we can leave it behind.
Now it has a name Keep one promise.
If 8 is present, it stays in the part we keep. That repeating promise is called an invariant.
No pressure. Just try it.Would this left-or-right shortcut work on an unsorted row?
1 / 4 A small step before this topic
Let’s picture it Keep a small part of a row.
Our budget is 5. The values 2 and 3 fit together.
One little change One more makes it too much.
Add 4: the sum is 9. Remove from the left until it fits again.
Now it has a name That moving part is a window.
left edge
kept values
right edge
With nonnegative values, adding cannot lower the sum. That predictable change tells us which boundary to move.
No pressure. Just try it.Could adding a negative number make a too-large sum fit?
1 / 4 A small step before this topic
A familiar idea A window meets a negative number.
Find stretches totaling 2 in [2, −1, 1]. Adding −1 shrinks the sum, so the usual window direction is no longer reliable.
One new wrinkle Keep a running total instead.
Before anything: 0. After each number: 2, 1, 2. These totals describe the boundaries between items.
Try a tiny example Subtract two boundaries.
earlier: 0
current: 2
difference: 2
A current total of 2 minus an earlier total of 0 gives a stretch totaling 2.
Notice what matters Ask for the missing earlier total.
At total s, look for earlier total s − k. Count how often it appeared, then record s.
Make it reusable Prefix sums meet a dictionary.
The two valid stretches are [2] and [2, −1, 1]. We found both without trying every starting point again.
No pressure. Just try it.Why count earlier totals instead of keeping only a set?
1 / 6 A small step before this topic
Let’s picture it Open a box.
There’s a smaller box inside. We already know how to open a box.
One little change Do the same small job again.
Open the smaller box. Keep going until there isn’t another box inside.
Now it has a name That’s the idea of recursion.
A function asks itself to handle a smaller case. The empty box is our stopping point: the base case.
No pressure. Just try it.What if every box contains another box forever?
1 / 4 A small step before this topic
Let’s picture it A question can make two more.
Climb one or two stairs at a time. Four stairs can lead to a smaller question with three or two left.
One little change Look, the same question returns.
ways(3) also asks for ways(2). We’ve reached that smaller question twice.
Now it has a name Notice the repeated work.
Draw the calls to spot repeated questions. Later, we’ll save their answers instead of solving them again.
No pressure. Just try it.Do two calls with the same input always need separate work here?
1 / 4 A small step before this topic
Let’s picture it Order can expose structure.
Unsorted values [4,1,3,1] hide equal neighbors and make smallest-first processing awkward.
One little change Sort once, scan simply.
After sorting: [1,1,3,4]. Duplicates touch, boundaries become visible, and two pointers become possible.
Now it has a name Account for the price.
sort cost
simpler scan
reuse order
Comparison sorting usually costs O(n log n). It is worthwhile only when the simpler scan or repeated queries repay that cost.
No pressure. Just try it.Does sorting preserve original positions automatically?
1 / 4 A small step before this topic
Let’s picture it Count a tiny family of dots.
One parent has two children. Each child says, “There’s one of me.”
One little change Let the answers travel up.
The parent adds the two answers, then counts itself: 1 + 1 + 1.
Now it has a name That works for bigger trees too.
my answer
left answer
right answer
Ask each child about its little tree. Combine the answers. You don’t need to hold the whole tree in your head.
No pressure. Just try it.Do we also count the parent itself?
1 / 4 A small step before this topic
Let’s picture it Come back to our little map.
Home, park, café. Each place is a dot; each allowed walk is a link.
One little change A dot can also mean a situation.
“At park with a key” may allow a move that “at park without a key” cannot.
Now it has a name Choose dots that tell the whole story.
situation
legal move
new situation
A graph organizes possible moves. Sometimes places are enough. Sometimes a place plus a small extra fact is needed.
No pressure. Just try it.Must graph nodes always represent physical places?
1 / 4 A small step before this topic
Let’s picture it Try the nearest places first.
From home, find everything one road away before looking two roads away.
One little change Put new places at the back.
A queue keeps the earlier discoveries in front. That lets us finish one layer before the next.
Now it has a name That’s breadth-first search: BFS.
With equal-cost roads, the first layer reaching a place uses the fewest roads. Unequal costs need more care.
No pressure. Just try it.One road costs 100 minutes; two others cost 1 each. Fewest roads means fastest?
1 / 4 A small step before this topic
Let’s picture it Follow one path for a while.
Instead of finishing a whole layer, keep walking down one branch. This is depth-first search, or DFS.
One little change Remember where you’ve been.
For reachability, a seen set helps avoid walking around the same cycle forever.
Now it has a name But ask what “seen” means.
this attempt
undo mark
next attempt
For routes that cannot repeat a place, a place used in one attempt may be free in another. Clear path-only marks when returning.
No pressure. Just try it.Does “used in this path” always mean “forbidden in every path”?
1 / 4 A small step before this topic
Let’s picture it Socks before shoes.
Some jobs must wait for other jobs. Draw an arrow from the earlier job to the later one.
One little change Start with a ready job.
Do a job with nothing unfinished before it. Remove it from the waiting list and look for the next ready job.
Now it has a name That makes a dependency order.
A waits for B
B waits for A
Its name is topological order. A cycle means some jobs keep waiting on one another, so no such order exists.
No pressure. Just try it.A waits for B, and B waits for A. Can either start under these rules?
1 / 4 A small step before this topic
Let’s picture it Try a snack combination.
Pick apple first. See what can go with it. We’re exploring, not committing forever.
One little change Put it back and try again.
After exploring the apple choices, undo that pick. Now start with banana.
Now it has a name That’s backtracking.
Choose, explore, undo. Undoing keeps the next attempt from accidentally inheriting the last attempt’s choices.
No pressure. Just try it.Before a sibling attempt, what happens to path-only choices?
1 / 4 A small step before this topic
Let’s picture it We’ve answered this before.
route A
3 stairs left
route B
Two routes leave us with three stairs left. We may climb one or two at a time.
One little change Save these three little routes.
The routes are 1+1+1, 1+2, and 2+1. Count them once; both arrivals can reuse the answer.
Now it has a name That sharing is the heart of DP.
same question
reuse answer
Dynamic programming saves answers to smaller questions. Share only when the remaining question really is the same.
No pressure. Just try it.One route has a key left and the other doesn’t. Always the same remaining question?
1 / 4 A small step before this topic
Let’s picture it Two ways to fill a notebook.
One way: when a question appears, solve it if needed, then write down the answer.
One little change Or prepare the small answers first.
Fill answers in an order where each needed earlier answer is already available.
Now it has a name Same idea, different order.
Saving answers on demand is memoization. Filling them in a planned order is tabulation.
No pressure. Just try it.When filling a table, should an answer be ready before another cell uses it?
1 / 4 A small step before this topic
A familiar idea Give three places their own switches.
Rightmost is A, middle is B, leftmost is C. A lit bit means that place has been visited.
One new wrinkle Visit A, then C.
Turn on A’s bit: 001. Then C’s bit: 101. The same places in another order produce the same mask.
Try a tiny example Use OR to turn a bit on.
mask
single bit
OR combines
1 << v makes a single lit bit at position v. mask | (1 << v) includes that place without switching other bits off.
Notice what matters Use AND to ask whether it is on.
membership: &
include: |
all: 111
mask & (1 << v) is nonzero exactly when v is included. All three places visited gives 111.
Make it reusable Small notation. Still many possible sets.
Three bits allow eight masks. n bits allow 2ⁿ. A compact representation does not make every subset search cheap.
No pressure. Just try it.Does 101 remember whether A or C was visited first?
1 / 6 A small step before this topic
Let’s picture it One integer can hold many booleans.
Bits 101 mean switches A and C are on while B is off.
One little change Each operator asks a different question.
OR set
AND test
XOR toggle
shift
OR turns bits on, AND tests or keeps them, XOR toggles differences, and shifts move bit positions.
Now it has a name Mask before interpreting.
unbounded integer
explicit width
Python integers do not have a fixed signed width. Define width explicitly when simulating 32-bit behavior.
No pressure. Just try it.What does x & (1 << i) test?
1 / 4 A small step before this topic
Let’s picture it Equal values cancel.
x XOR x is zero, and x XOR 0 is x. Pairing equal values removes them regardless of order.
One little change One unpaired value remains.
In [4,1,4], XORing everything cancels the two 4s and leaves 1.
Now it has a name The guarantee is the algorithm.
pairs cancel
one odd remains
This works because every other value occurs exactly twice. Different multiplicities require different state.
No pressure. Just try it.Would ordinary XOR find one unique value if others appear three times?
1 / 4 A small step before this topic
Let’s picture it XOR also has prefix cancellation.
prefix before left
range
prefix after right
Let prefix[i] XOR the values before i. A range left..right equals prefix[right+1] XOR prefix[left].
One little change The shared prefix disappears.
Everything before left appears twice in that XOR and cancels bit by bit.
Now it has a name Updates make prefixes stale.
static queries
dynamic structure
This is ideal for many queries on frozen data. Online changes need a Fenwick or segment tree using XOR combine.
No pressure. Just try it.Which operation replaces subtraction for XOR prefixes?
1 / 4 A small step before this topic
Let’s picture it Remainders make numbers smaller.
gcd(48,18) equals gcd(18,12), because 48 = 2·18 + 12.
One little change Common divisors are preserved.
A number divides both 48 and 18 exactly when it divides 18 and their remainder 12.
Now it has a name Repeat until remainder zero.
gcd(18,12)→gcd(12,6)→gcd(6,0). The remaining nonzero number is the gcd.
No pressure. Just try it.Why may a be replaced by a mod b?
1 / 4 A small step before this topic
Let’s picture it A composite has a smaller prime factor.
2 multiples
3 multiples
5 multiples
Mark multiples of 2, then 3, then 5. Unmarked values remain prime.
One little change Start crossing at p squared.
2p done
3p done
p² starts new
Smaller multiples of p already had a smaller factor and were crossed out earlier.
Now it has a name Stop base primes at square root.
small factor
×
other factor
Any composite up to n has at least one factor no larger than √n.
No pressure. Just try it.Why can crossing multiples of p begin at p²?
1 / 4 A small step before this topic
Let’s picture it An exponent can halve.
x¹³ = x·(x²)⁶. Squaring the base and halving the exponent skips repeated multiplication.
One little change Read exponent bits.
bit 1 → multiply
square
shift
When the current exponent bit is one, multiply the answer by the current base. Square the base every step.
Now it has a name Reduce throughout modular work.
reduce base
multiply
reduce again
(a·b) mod m equals multiplying their residues then reducing. This controls size and preserves the remainder.
No pressure. Just try it.How many squaring steps does exponent n need?
1 / 4 A small step before this topic
Let’s picture it Some recurrences are linear transitions.
Fibonacci keeps [F(n),F(n−1)]. One fixed 2×2 matrix advances that state by one step.
One little change Many equal steps can be squared.
Applying transition T eight times is T⁸. Binary exponentiation squares matrices just like numbers.
Now it has a name State dimension controls cost.
A d×d multiplication costs O(d³) naively. This helps when d is small and n is huge.
No pressure. Just try it.What property lets powers of one transition skip many steps?
1 / 4 A small step before this topic
Let’s picture it Order may or may not matter.
Choosing 2 people from 5 gives 10 groups. Arranging 2 roles from 5 gives 20 ordered assignments.
One little change Build binomial coefficients safely.
multiply next factor
divide by k
C(n,k)=C(n,k−1)·(n−k+1)/k. Divide at every step; the result stays integral.
Now it has a name State the counted objects first.
what differs?
does order matter?
Permutations, combinations, repeated choices, and indistinguishable items have different formulas.
No pressure. Just try it.Does C(n,k) distinguish the order of selected items?
1 / 4 A small step before this topic
Let’s picture it Three points reveal a turn.
From A to B to C, a cross product is positive for a left turn, negative for right, and zero for collinear.
One little change Sign often matters more than angle.
Convex-hull construction needs to know whether the boundary turns the allowed way, not the actual angle in degrees.
Now it has a name Sorting gives a walk around candidates.
sorted points
lower hull
upper hull
Sort points lexicographically, build lower and upper chains, and remove middle points that make the wrong turn.
No pressure. Just try it.What does a zero cross product mean?
1 / 4 A small step before this topic
Let’s picture it The stream length is unknown.
unknown length
one saved item
Items arrive one by one and cannot all be stored. We still want each final item equally likely.
One little change New item i gets chance 1/i.
replace: 1/i
keep: (i−1)/i
If selected, it replaces the reservoir. Otherwise the current sample stays.
Now it has a name Old items keep equal probability.
An earlier item survives step i with probability (i−1)/i, turning its prior 1/(i−1) chance into 1/i.
No pressure. Just try it.Must the total stream length be known beforehand?
1 / 4 Keep a connection for later.
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