The Lumenary · K–2

Following and giving precise instructions

Name: ______________________

Date: ______________________

You will learn to

Follow a sequence of movement instructions on a grid and predict where it ends
Write a sequence of instructions to reach a target, and test it
Find and fix a mistake (a "bug") in a sequence of instructions
Explain why the order of the steps can change the result
Explain why instructions must be precise enough to follow without guessing, and that there is often more than one correct program

1.
You start on the star. Follow the program → → ↑. Where do you end?
- A.2 squares right and 1 square up
- B.1 square right and 2 squares up
- C.3 squares right
- D.2 squares right and 1 square down
2.
The tree is 4 squares to the right of the star, in the same row. How many → cards do you need to get there?
3.
The mango is 1 square right and 2 squares up from the star. Which programs get you there? Choose all the correct ones.
- A.↑ ↑ →
- B.→ ↑ ↑
- C.↑ → ↑
- D.→ → ↑
- E.↑ →
4.
Ayesha wants to reach the square 2 right and 2 up. Her program is → ↑ ← ↑. She tests it and misses. Which card is the bug?
- A.The 1st card (→)
- B.The 2nd card (↑)
- C.The 3rd card (←)
- D.The 4th card (↑)
5.
A floor robot faces up the page. Program A is forward, turn right. Program B is turn right, forward. Do they end on the same square?
- A.No, they end on different squares
- B.Yes, they end on the same square
6.
Gideon's game: true or false? "There is only one correct program to reach a square."
- A.True
- B.False
7.
Which instruction is precise enough for a robot to follow without guessing?
- A.Move 3 squares forward
- B.Go over there
- C.Walk forward a bit
- D.Go the usual way
8.
What is the smallest number of arrow cards that can take you from the star to the square 3 right and 2 up?
9.
Margaret's puzzle: how many different shortest programs go from the star to the square 2 right and 1 up?

Answer key — Following and giving precise instructions

1.
→ moves one square right, → moves another square right, and ↑ moves one square up. You end 2 squares right and 1 square up from the star.
2.
Each → card moves one square to the right, so to move 4 squares you need 4 cards: → → → →.
3.
Any program with one → and two ↑ reaches the mango: ↑ ↑ →, → ↑ ↑ and ↑ → ↑ all work. There is more than one correct program! → → ↑ goes too far right, and ↑ → is not high enough.
4.
Following → ↑ ← ↑ ends 0 squares right and 2 up — the ← undid the →. The 3rd card is the bug. Changing it to → gives → ↑ → ↑, which ends 2 right and 2 up.
5.
In A the robot moves forward while facing up, so it moves one square up, then turns. In B it turns to face right first, so it moves one square right. Different squares — the order of the steps changes the result.
6.
False. For example, ↑ → → and → → ↑ both reach the square 2 right and 1 up. That one example is a counterexample. Usually there are many correct programs.
7.
Move 3 squares forward is precise: it says what to do (move forward) and exactly how far (3 squares). The others need guessing: where is "there", how much is "a bit", what is "the usual way"?
8.
Each card moves you one square, either across or up/down. You need 3 squares across and 2 up, so at least $3 + 2 = 5$ cards, and → → → ↑ ↑ shows that 5 is enough. The Lean theorem taxi_le_length proves that no program can be shorter than "across plus up".
9.
A shortest program has two → and one ↑. The ↑ can be first, second or third: ↑ → →, → ↑ →, → → ↑. That makes 3 shortest programs. Margaret's computer finds the same, and 20 for the square 3 right and 3 up.