NACLO 2025 - Problem KGoing By the Numbers

Invitational round | 15 points | 69.23% | Problem statement | Official solution | Tags: Computational

K1 is straightforward: past tense.

K2 is basic vector product:

$$\begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 2 \end{bmatrix}\cdot \begin{bmatrix} 7 & 2 & -2 & -5 & -10 & 1 & 4 \end{bmatrix} = 0 + 0 + 0 + 0 - 10 + 1 + 8 = -1$$

K3 is a bit tedious, but we don't have to do literally every multiplication. Observe:

• w1 returns positive iff the first component is negative, which are all plural nouns.
• w2 returns positive iff the fourth component is positive, which are all past tense verbs.
• w3 returns positive iff the second component is positive, which are all animate objects.
• w4 returns positive iff x5 + x6 + 2 * x7 > 0, containing: apple, oranges, pizza, salads — all food items.
• w5 returns positive iff the fourth component is negative, which are all nouns.
• w6 returns positive iff x1 - 2 * x7 > 0, containing: apple, electrician, interpreter. These are all singular nouns starting with a vowel (i.e., uses "an" instead of "a").
• w7 returns positive iff the third component is positive, containing: mug, plates, sticks, television. These are all fragile items.
• w8 returns positive iff x1 - x7 > 0, containing: apple, book, electrician, interpreter, mug, pizza, purse, squirrel, restaurant, television. These are all singular nouns.
• w9 returns positive iff the sixth component is negative, containing: book, envelopes, purse, restaurant. These are things you can "open" (but not "put in" because "book" is not a container).

For (a), (b), (c), we need to choose from w2, w3, w4. (a) selects an animate object, so w3. (b) selects a food, so w4. (c) selects a past tense verb, so w2. For (d), any noun should work, so for example: "I like the ____". For (e), as already said, we need the context of "an ____". For (f), it should select a fragile item: "I broke the ____". For (g), it should select a singular noun, so for example: "I like this ____". For (h), as already said, the only viable context is "I opened the ____".

K4 is again very tedious:

1. Positive: w5, w7, w8, w9. So it's a singular noun, inanimate, not food, doesn't start with a vowel, can be opened, and is fragile. For example: cupboard.
2. Positive: w1, w5. So it's a plural noun, inanimate, not food, doesn't start with a vowel, can't be opened or broken. For example: planets.
3. Positive: w4, w5, w7, w8. So it's a singular noun, inanimate, food, doesn't start with a vowel, can't be opened, and is fragile. For example: cookie.