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Subjects · Leaving Cert Applied Maths

Leaving Cert Applied Maths: Dynamic programming & Bellman

How often Dynamic programming & Bellman comes up on the Applied Maths papers, every year it was asked, and questions to try.

HL Asked on 2 of the last 3 Higher Level papers, most recently in 2025. most years

OL Asked on 1 of the last 3 Ordinary Level papers, most recently in 2024.

Dynamic programming & Bellman, Higher Level(6 marks)

Quick ones on Dynamic programming & Bellman.

(a)What does Bellman's principle of optimality say?
  1. The cheapest next step is always the best one to take
  2. Every remaining part of an optimal route is itself optimal
  3. Every vertex must be visited exactly once on the route
(b)In a dynamic programming (Bellman) table, you usually work in which direction?
  1. In increasing order of edge weight
  2. Forwards from the start, greedily
  3. Backwards from the final stage
(c)In dynamic programming, what is a stage?
  1. A step at which a decision is made
  2. The final destination
  3. A vertex with the lowest value
Show the answers

(a) Every remaining part of an optimal route is itself optimal

(b) Backwards from the final stage

(c) A step at which a decision is made

Your turn: Higher Level questions on Dynamic programming & Bellman.

Higher Level

Asked on 2 of the last 3 Higher Level papers, most recently in 2025. most years

Every paper, year by year

YearWhere it came up
2025Q6
2024Q8
2023Not asked

Links open the State Examinations Commission’s paper for that year.

More Dynamic programming & Bellman questions

Dynamic programming & Bellman, 2 marks

A machine can be kept or replaced at the start of each year. In the dynamic programming model, these are the…?

  1. States
  2. Stages
  3. Actions
Show the answer

Actions

The years are the stages, the machine's age is the state, and keep or replace are the actions. Each action has a cost and moves you to a new age in the next stage.

Dynamic programming & Bellman, 3 marks

Why does working backwards through the stages save work in dynamic programming?

  1. It can skip the last stage of the network
  2. Each state's best value is found once and reused
  3. It only follows the cheapest arcs backwards
Show the answer

Each state's best value is found once and reused

Once you know the best value from every state in a stage, earlier stages just add one arc to it. You never have to list and total every complete route.

Dynamic programming & Bellman, 3 marks

A staged network offers 3 choices at each of 4 decisions. How many complete routes are there?

  1. 81
  2. 12
  3. 64
Show the answer

81

Routes multiply: 3 × 3 × 3 × 3 = 3⁴ = 81. Bellman's backward pass avoids totalling all 81; it compares only a few options at each state.

Ordinary Level

Asked on 1 of the last 3 Ordinary Level papers, most recently in 2024.

Every paper, year by year

YearWhere it came up
2025Not asked
2024Q5
2023Not asked

Links open the State Examinations Commission’s paper for that year.

More Dynamic programming & Bellman questions

Dynamic programming & Bellman, 2 marks

Dynamic programming solves a staged network by working…

  1. Forwards, choosing the cheapest edge each time
  2. At random through the stages
  3. Backwards from the final stage
Show the answer

Backwards from the final stage

You start at the destination and find the best value from each state to the end, stage by stage, until you reach the start.

Dynamic programming & Bellman, 2 marks

Can dynamic programming find the route with the greatest total profit?

  1. Only if all profits are equal
  2. Yes, take the largest value at each state
  3. No, it only finds shortest distances
Show the answer

Yes, take the largest value at each state

The backward method works the same way for a maximum: at each state choose the action giving the biggest value to the end.

Dynamic programming & Bellman, 3 marks

Bellman's principle of optimality says that…

  1. The rest of an optimal route is also optimal
  2. The shortest edge is always in the best route
  3. The first choice alone decides the best route
Show the answer

The rest of an optimal route is also optimal

Whatever state you reach, the remaining decisions of an optimal policy must be optimal from that state. This is why working backwards gives the best overall route.

Other Applied Maths topics

All of Leaving Cert Applied Maths