AMC 10 · 2017 · #22
Grade 10 probabilityPick an answer.
The walk has no time limit, so there is nothing finite to list. The fix is to name the answer from each starting spot as an unknown. Because every move averages over 8 neighbours, each unknown equals the average of the neighbouring unknowns, which turns an endless process into a small system of equations. The square's symmetry first shrinks 9 interior spots to 3 families, so the system has only 3 unknowns.
Draw the board
The board has an inside and a boundary.
A wall one step away cannot be leapt over by steps of size one, so the walk must land on it.
10.S-CP.A.1Draw A DiagramFold nine spots into three
Symmetry folds nine spots into three.
Turning the whole picture a quarter turn changes nothing, so spots that turn onto each other must carry the same number.
Turning the whole picture a quarter turn changes nothing, so spots that turn onto each other carry the same number.
▸ Why?
A rotation carries the board onto itself without stretching, so the rules look identical afterwards.
▸ Why?
Four quarter turns return everything to its start, so the spots fall into small closed families.
Name one unknown per family
Each kind gets one unknown.
Where you stand is worth exactly the average of the eight places you might stand one second later.
10.S-CP.B.7Introduce A VariableEquation at the origin
The centre gives the first equation.
From dead centre the game cannot end yet, so the origin is just the midpoint between the two other families.
9.A-CED.A.3Identify SubproblemsEquation at an axis point
An axis spot gives the second.
Facing the flat middle of a wall, three of the eight steps end the walk with no corner at all.
9.A-CED.A.3Identify SubproblemsEquation at a diagonal point
A diagonal spot gives the third.
A corner has only one inside neighbour, so all corner luck has to enter through the diagonal spots.
9.A-CED.A.3Identify SubproblemsSolve the three equations
Solving the system gives 4/35.
Two equations pin b and a as multiples of c, and the third then has only one unknown left.
9.A-REI.C.6Convert To AlgebraRead off m + n
Its parts add to 39, choice (A).
The starting square is the origin, so the origin's unknown is the one the question wants.
6.NS.B.4Convert To AlgebraWhen a walk has no time limit, name the answer from each spot as an unknown: every spot is worth the average of its neighbours, and symmetry shrinks nine unknowns down to three easy equations.
- Draw the board
- Fold nine spots into three
- Name one unknown per family
- Equation at the origin
- Equation at an axis point
- Equation at a diagonal point
- Solve the three equations
- Read off m + n