Q4 — Games: Strategy + Alpha-Beta

Backups give root 53 via B,D; alpha-beta prunes G,H,K,L; one leaf change (E=98) forces the MinMax value to 98.

The Question

The figure shows a 3-ply game tree with evaluation function values defined at horizon. The nodes in the horizon are labeled from A to L. Use these labels when asked to enter a horizon node or a list of horizon nodes.

Tie-breaker: when several nodes qualify then select the left most node, if tie persists then select the deepest node among the left most nodes.

#	Question	Official answer
4.1	Which of the following is a strategy for MAX?	`C`
4.2	List the horizon nodes in the best strategy for MAX (ascending)	`B,D`
4.3	List all the horizon nodes pruned by Alpha-Beta	`G,H,K,L`
4.4	Force the MinMax value to 98 by changing the eval of only one horizon node	`E,98`

The Topic in 60 Seconds

Backup: MAX squares take max of children; MIN circles take min.
Strategy for MAX: one committed move at every MAX node — here root picks ONE MIN subtree and each surviving MAX square picks ONE leaf (2 leaves total).
Alpha-beta: a finished child that already fails the window kills the rest of the branch.
Force MinMax: find which top-level subtree must move, then push its blocking MAX value.

Deep-dive: Game Strategies, Force MinMax Value, Alpha-Beta Pruning.

Step 0 — Back up every leaf

MAX square	Computation	Value
$L2_1$	$\max(18, 69)$	69
$L2_2$	$\max(30, 53)$	53
$L2_3$	$\max(15, 27)$	27
$L2_4$	$\max(98, 14)$	98
$L2_5$	$\max(19, 37)$	37
$L2_6$	$\max(61, 68)$	68

MIN circle	Computation	Value
$L1_1$	$\min(69, 53)$	53
$L1_2$	$\min(27, 98)$	27
$L1_3$	$\min(37, 68)$	37

Root $= \max(53, 27, 37) = \mathbf{53}$ via $L1_1$ .

4.1 — Strategy for MAX → `C`

A legal strategy commits the root to exactly one MIN child, and then names exactly one leaf under each MAX grandchild beneath it — two leaves overall. Options shaped like "two leaves from different subtrees of one MIN" fit; options mixing leaves from two different MIN subtrees (impossible for one root move) or listing four-plus leaves do not. The option with this shape is C ✓

4.2 — Best strategy leaves → `B,D`

The guarantee is maximized through $L1_1$ (53). Its MAX squares pick their best leaves: $L2_1 \to$ B ( $\max(18,69)=69$ ), $L2_2 \to$ D ( $\max(30,53)=53$ ). Guarantee $= \min(69, 53) = 53 =$ game value ✓

4.3 — Alpha-beta prunes → `G,H,K,L`

Left-to-right traversal:

Node	Window	What happens
$L1_1$ (MIN)	$(-\infty, \infty)$	$L2_1$ : A=18, B=69 → returns 69. Then $L2_2$ : C=30, D=53 → returns 53. $L1_1 = 53$ , root α = 53
$L1_2$ (MIN)	$(53, \infty)$	$L2_3$ : E=15, F=27 → returns 27 ≤ 53 → α-cutoff: $L2_4$ never opened → G, H pruned
$L1_3$ (MIN)	$(53, \infty)$	$L2_5$ : I=19, J=37 → returns 37 ≤ 53 → α-cutoff: $L2_6$ never opened → K, L pruned

Pruned horizon nodes: G, H, K, L ✓ (no pruning inside $L1_1$ — both its squares complete because no bound is active before the first subtree finishes.)

4.4 — Force MinMax to 98 → `E, 98`

Root $= \max(L1_1, L1_2, L1_3)$ . To reach 98, some subtree must rise to 98 while the others stay ≤ 98.

Raising $L1_1$ or $L1_3$ to 98 needs both of their MAX squares ≥ 98 — impossible with one leaf change ( $\min$ structure caps them).
$L1_2 = \min(L2_3, L2_4)$ : $L2_4 = \max(98, 14) = 98$ already; so $L1_2 = 98$ requires only $L2_3 = \max(E, F) = 98$ → set E = 98 (F works identically; E is leftmost per the tie-breaker).

Check: $L2_3 = 98 \Rightarrow L1_2 = \min(98, 98) = 98 \Rightarrow$ Root $= \max(53, 98, 37) = \mathbf{98}$ ✓

98 is also the largest usable value (the biggest eval on the tree) — any target above it would be unreachable at all.

Watch the whole tree resolve

1 / 5

Leaves A-L carry static evaluations. Nothing backed up yet.

backing upevaluated / backed uppruned by alpha-betabest line / force point

Tricks & Tips

Answers: 4.1 C · 4.2 B,D · 4.3 G,H,K,L · 4.4 E,98

The Question

The Topic in 60 Seconds

Step 0 — Back up every leaf

4.1 — Strategy for MAX → C

4.2 — Best strategy leaves → B,D