Working Backwards
When a problem hands you the ENDING and hides the START, try undoing the chain yourself first — then watch how starting at the end cracks it open.
1 Explore — try these first
Try before you watch. Pick a level below and give the problem an honest try on paper first — wrong turns and all. Then open the video to see the trick. Every level rides one habit: When a problem describes a chain of actions and gives you the final state, undo each step in reverse order — last step first, every operation flipped to its inverse.
★ · Reverse the Operations
Sarah spent half of her allowance on a book. Then she spent $5 on lunch. She has $3 left. How much allowance did she start with?
L0 · Reverse the Operations
Ava and Ben are sorting welcome cards. Ava gives 18 cards to Ben. After that, both Ava and Ben have 47 cards. How many cards did Ava have at the start?
L1 · Reverse the Operations (multi-op)
A helper doubles her stack of welcome notes by adding an equal pile. Then she gives away 6 notes. After both steps she has 18 notes. How many notes did she start with?
L2 · Reverse Transfers, Track Everyone
Three volunteers sort crayons. First Mira gives 6 crayons to Jay. Then Sol gives 4 crayons to Mira. After that, all three volunteers have 20 crayons each. How many crayons did Mira have at the start?
L3 · Find the Invariant First
On Leo's 14th birthday, Amir is 41 years old — notice 14 and 41 use the same two digits reversed. The NEXT time Leo's age is the digit-reverse of Amir's age, how old will Leo be?
L4 · Reverse the Sum, Not the Parts
Two food pantries each start with a positive whole number of meal boxes. The same three steps happen to each: 10 boxes are added, then the total is doubled by a sponsor, then 30 boxes are sent out. After all three steps, the two pantry counts add up to 130. How many possible starting counts could Pantry A have had?
L5 · Constant Speeds, Proportional Distances
Two couriers start at the same time from opposite ends of a hallway and walk back and forth at constant but different speeds. They first meet 30 m from the welcome desk. Each turns once at the ends, then they meet again, this time 20 m from the medicine room at the other end. How long is the hallway?
L6 · Short List, Then Filter
Two couriers start at the same time from opposite ends of a hallway and walk back and forth at constant but different speeds. Their first meeting is 27 m from the welcome desk, and their third meeting is 15 m from the welcome desk. The faster courier walks 2 m/min more than the slower one, and both speeds are whole numbers of metres per minute. How long is the hallway?
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
★ · Reverse the Operations
Peek the trick — Undo the chain, last step first
Every forward action has an inverse: add becomes subtract, double becomes halve. To find the start, walk the steps backward from the known end and apply each inverse in reverse order.
L0 · Ava's Card Stack
Peek the trick — Start at the end, undo one move
When a story gives the final state, work backwards from it. A gift you made becomes a take-back when you reverse it.
L1 · Notes That Doubled
Peek the trick — Two undos, reverse order
With several operations chained, invert each one and undo them in reverse order. Doubling becomes halving, giving away becomes adding back — and the LAST step is undone first.
L2 · Three Volunteers & Crayons
Peek the trick — Reverse each transfer, watch every name
When several people pass items along and end at known counts, reverse each transfer in reverse order. In every undo, only the two people in that exchange change — everyone else stays the same.
L3 · Amir and Leo
Peek the trick — Lock onto the thing that never changes
When the chain is a relationship instead of arithmetic, find the quantity that never changes — here the age gap. For two-digit ages that are digit-reverses, the gap is always 9 times the difference of their digits.
L4 · Two Pantries
Peek the trick — Undo the total in one equation
When the same chain hits two values and only the SUM of the results is given, work backwards on the sum in a single equation — not on each value separately. That turns the problem from solving into counting.
L5 · Two Couriers
Peek the trick — Track combined distance between meetings
Two walkers at steady speeds cover distance in proportion to time. At their first meeting their combined distance is one hallway length; by the second meeting it is three lengths, so each walker's own distance triples too.
L6 · Couriers, Three Meetings
Peek the trick — List the few candidates, then filter to one
When the geometry leaves only a few possible answers, list every candidate, then use an extra constraint — integer speeds, a fixed speed gap — to pick the single one that survives.
3 Master — practice on your own
Print the practice sheet and solve without the videos. Check your answers at the back — if one is wrong, the answer key names the trick so you know exactly which video to rewatch.
Download fresh practice problems PDF
