Check intermediate results in a lattice-target investigation

The investigation brief, ungraded

A marker starts at the origin and takes n equal steps, choosing east,north,west or south independently and uniformly each time. A circular target is centred at the origin. Investigate how endpoint capture changes with step count,target radius and step length. Formulate a precise question, obtain exact small-case results, run your own simulation, test a conjecture and communicate justified conclusions. The complete written investigation is ungraded

Plan the investigation before using the models

Use three phases: formulation, exploration and communication. First record a conjecture and the quantities you will vary. Keep your initial and revised ideas, code or spreadsheet, input ranges, raw or summarised results, plots and checks. Attempt each written task before reading its model

Formulate a precise question, ungraded

Choose an endpoint-capture question for n=1..6 and at least three radii. Define the outcome, randomisation, units, target boundary and quantities held fixed. Distinguish capture at the final time from ever entering the target or staying inside throughout. Explain why an abstract marker need not describe a real navigation device

Model formulation and assumptions

Let the direction vectors be independent uniform choices among the four unit compass directions. The endpoint is their sum multiplied by a positive step length. Capture includes the circular boundary. Study how its probability depends on the step count and the ratio of target radius to step length. Fix the start at the origin, step timing and direction law. Ignore obstacles, inertia and measurement error. These assumptions define a mathematical experiment; they are not established facts about a real device

How many ordered direction sequences are there for four steps if each step can be E, N, W or S?

Use vectors or complex numbers for the same endpoint

Represent east,north,west,south by s,si,−s,−si. Adding these complex steps gives z=x+yi and |z|²=x²+y². For E,N,W,N with s=2,the endpoint is4i,or(0,4),with squared distance16. The path order does not change its sum,but different ordered paths remain different random outcomes

A path starts at the origin and takes steps E, N, W, N, each of length 2 units. Give the squared distance of its endpoint from the origin

Does the shown four-step path stay inside the unit circle throughout?

Count paths rather than distinct endpoints

After two steps there are16 ordered sequences but only nine endpoints. Four paths return to the origin:EW,WE,NS,SN. Each diagonal(±1,±1) has two orders. Each axial distance2 endpoint has one path. Thus the endpoint distribution is not uniform over its nine points

What is the two-step probability of ending at the origin?

Starting at the origin, a path takes two unit steps, each E, N, W or S. How many ordered paths end at (1,1)?

A circle is a test on each endpoint

With unit steps and R=√2,capture means x²+y²≤2. After two steps,the origin contributes4 paths and the four diagonal boundary points contribute2 each. Therefore N(A)=12 and p=12/16=3/4. A strict inequality would exclude those diagonal points and give1/4 instead

Starting at the origin, take two independent unit steps, each uniformly E, N, W or S. Give the probability that the endpoint lies within or on the circle of radius √2

Use a recurrence to count exactly

Let Cⱼ(x,y) count length-j paths ending at(x,y) in unit-step coordinates. Set C₀(0,0)=1 and all other entries0. The final step must come from one of four neighbours,so sum their previous counts. Use a new table at each time step. The counts sum to4ʲ

Justify the count and reachability tests, ungraded

Prove the recurrence by a disjoint,exhaustive partition. Prove that(x,y) is reachable after n unit steps exactly when |x|+|y|≤n and n−|x|−|y| is even. Use these facts to check your table. Do not present a few matching outputs as a general proof

Model proof: use the last step and reversible pairs

Every path has one final direction. Removing it gives exactly one path to the corresponding neighbour; appending it reverses that operation. The four direction classes are disjoint and exhaustive,proving the recurrence. A path needs at least |x|+|y| steps,and each step changes coordinate-sum parity. Conversely,take a shortest coordinate path and append (n−|x|−|y|)/2 east–west pairs. These pairs add two steps and zero displacement,so every point meeting both conditions is reachable

Worked exact comparison and a revised conjecture

For a target of radius 1 and unit steps, the probabilities for the first four step counts are shown below. At odd times the origin is unreachable, while at even times it is the only reachable point within that radius. The nearest four endpoints after three steps each have 9 paths. Capture therefore need not decrease at every extra step. The final line gives the four-step count and probability for a target of radius √2.

Starting at the origin, take four unit steps, each E, N, W or S. How many ordered paths end within or on the circle of radius √2?

Implement two exact methods, ungraded

Write and execute both an ordered-path enumerator and a count-table recurrence for n=0..6. Compare every endpoint count,not just totals. Check total4ⁿ,nonnegative integer entries,symmetry,reachability and target counts. Record code or formulas and actual results. Independent implementation and execution are ungraded

Model algorithms and audit table

Enumeration: for each integer from0 to4ⁿ−1,decode exactly n base-four digits,keeping leading zeros; map digits0,1,2,3 to E,N,W,S and add the endpoint count. At n=0 there is one empty path at the origin. Recurrence: initialise {(0,0):1};for each step,create an empty next table and add each current count to its four neighbours,then replace current by next. Audit totals for n=0..6:1,4,16,64,256,1024,4096. Return-to-origin counts are1,0,4,0,36,0,400. Both methods should agree entry by entry

Starting at the origin, take six unit steps, each E, N, W or S. How many ordered direction sequences return to the origin?

Conditioning changes the denominator

For n=2,R=√2,let B be final x≥0. B includes6 paths on the vertical axis and5 with positive x,so N(B)=11. Eight paths satisfy both capture and B,so Pr(A|B)=8/11. Conversely Pr(B|A)=8/12=2/3. Symmetry does not give Pr(B)=1/2 because the axis has positive probability

After two independent uniform unit compass steps from the origin, let A mean that the endpoint is within or on the circle of radius √2 and B mean that its x-coordinate is nonnegative. Give Pr(A|B)

Run an independent simulation, ungraded

For n=4,R=√2,s=1,simulate at least400 independent walks,then a larger run. For each walk generate four fresh independent uniform direction codes,accumulate its endpoint and record the capture indicator. Save enough evidence to reproduce the run:tool and version,seed when supported or frozen direction records,input settings,trial count and result. Compare with33/64. Repeat for an odd n and explain any difference. Your technology execution and interpretation are ungraded

Model simulation procedure

Repeat N times:reset x=y=0;draw n fresh codes uniformly from0,1,2,3;update(x,y) by(1,0),(0,1),(−1,0),(0,−1),respectively;record H=1 if s²(x²+y²)≤R² and0 otherwise. Estimate p by ΣH/N. In a spreadsheet,use one row per walk,one recorded direction code per step,then endpoint and indicator columns. Freeze generated codes before comparing settings so recalculation does not silently replace the sample. Check each simulated endpoint against the reachability rule. A deterministic question seed is not your simulation record

Worked illustrative results, not claimed simulation output

Suppose a supplied illustrative record has 194 captures in 400 trials. A second illustrative record has 834 captures in 1600 trials. The estimates and their signed differences from the exact probability are shown below. These counts show the calculations; no learner execution or empirical experiment is claimed. Keep your own results even when they differ

An illustrative record has 194 captures in 400 trials, while the exact probability is 33/64. Give the absolute difference between the observed relative frequency and the exact probability

Explain sampling variability rather than promise agreement

For independent capture indicators with probability p,the sample proportion has mean p and variance p(1−p)/N,using the existing independent-sum rules. Increasing N reduces this variance,but a particular larger run can have a larger absolute error. Reusing one direction code for every step,forgetting to reset endpoints or counting distinct endpoints equally changes the experiment and creates systematic error

A larger run has a larger absolute error than a smaller run. What follows?

Change one assumption and evaluate, ungraded

Compare longer steps with a fixed physical radius,then scale the radius with the steps. Test at least two n values and explain the invariant result. Propose a biased-direction or dependent-step extension;state which equal-probability counting argument fails and what would replace it. Keep that extension distinct from the solved uniform model

Model scale comparison and a model boundary

For n=2,R=2,s=1,every endpoint is captured,so p=1. Keeping R=2 but taking s=2 means unit-step coordinates must satisfy u²+v²≤1;only the four returning paths qualify,so p=1/4. If R also doubles to4,the probability returns to1. Generally multiplying s and R by the same positive factor cancels from s²(u²+v²)≤R². With independent biased directions,ordered paths need product weights instead of1/4ⁿ;with dependence,the transition probabilities must depend on the relevant state. The uniform counts alone cannot supply those new probabilities

Starting at the origin, take two independent steps of length 2, each uniformly E, N, W or S. Give the probability that the endpoint lies within or on the circle of radius 2

Write and defend your conclusion, ungraded

Produce a report with your precise question and assumptions,initial conjecture,exact small cases,proof,independently executed algorithms and simulation,plots or tables,comparison,changed assumption and limitations. Distinguish your observed results from supplied illustrative values. State what can be concluded about the mathematical marker and what would need evidence for a real device. Written communication and assessment judgement are ungraded

Model report conclusion

For independent uniform unit compass steps,ordered paths are equally likely but endpoints are not. Last-step counting gives exact endpoint probabilities. At four steps and radius√2,the exact capture probability is33/64. Radius1 capture is not monotone in step count because parity changes the reachable points. Independent simulation should fluctuate around the exact probability;one run does not prove the probability or validate a physical model. Scaling both target and step length preserves capture,while increasing only step length cannot add captured endpoints. Bias,dependence,obstacles or a different path event require a revised model and new evidence

Teacher guidance and assessment boundary

Review the learner’s question and original conjecture before showing models. Inspect their own code or spreadsheet,stored draws,endpoint comparisons,proof and interpretation. Ask them to explain a failed conjecture and one implementation check. Judge the full report with the school’s criteria. The app grades only supplied-model calculations;it does not observe independent execution,award a SAC result or certify investigation mastery