This is a set of **independent estimation and probability brainteasers** of the kind asked rapid-fire from a slide deck. Treat each one separately. For every part, **state any assumptions you make** and show the back-of-the-envelope reasoning, not just the final number. ### Clarifying Questions to Ask For a rapid-fire brainteaser round like this, a few quick scoping questions show maturity before you dive in: - Should I **show my reasoning out loud**, or is only the final number expected? (For estimation problems the reasoning is usually the point.) - For the estimation parts, do you want an **order-of-magnitude** answer or a precise figure? - Am I allowed to **state and use assumptions** freely (e.g. movie length, frame rate, the Moon's angular size), or are specific inputs provided? - For the floor-tiling and storage parts, do you care about **real-world layout/waste** (cut tiles, overhead) or just the idealized arithmetic? - For the unit-heavy parts, which **convention** should I use — decimal vs. binary for "TB," and which definition of a "movie length"? ### What a Strong Answer Covers Across these brainteasers the interviewer is reading for general quantitative-reasoning signal, not memorized trivia: - **Unit discipline** — converting feet↔yards, seconds↔hours, bytes↔TB cleanly and consistently. - **Stating assumptions explicitly** before computing, and noting how the answer moves if they change (sensitivity). - **Order-of-magnitude instinct** — recognizing useful anchors (e.g. $2^{10} \approx 10^3$, small-angle approximation) and getting the magnitude right. - **Choosing the fast path** — spotting when a shortcut beats brute force, rather than grinding through every case. - **Physical/intuitive justification**, not just a number — especially for the conceptual parts (buoyancy, geometry). - **Sanity-checking** the result (enumeration, comparison to a known value, plausibility). - **Calm structure under rapid-fire pressure** — partitioning each problem and not getting flustered by the pace. --- ### 1) Tile a floor A floor measures **15 ft × 20 ft**. You cover it with square tiles, each of area **1 square yard**. Approximately **how many tiles** are needed? ```hint Units first Both dimensions and the tile are easiest in the **same unit**. Recall $1\ \text{yd} = 3\ \text{ft}$, so a 1-square-yard tile is $3\ \text{ft} \times 3\ \text{ft}$. What's that in $\text{ft}^2$? ``` ```hint Area vs. layout There are two defensible answers: a pure **area ratio** (floor area ÷ tile area, the material lower bound) and a **whole-tile layout** that rounds each dimension up to a whole number of tiles. Mention which one you mean — they differ here. ``` ### 2) Doubling account An account starts with **$1,000** and **doubles every year**. **How much is in the account after 20 years?** ```hint Doubling each year is geometric growth — after 20 years the multiplier is a power of two. Which power? The anchor $2^{10} = 1024 \approx 10^3$ lets you nail the order of magnitude fast. ``` ### 3) Estimate the Earth–Moon distance You are sitting by a window looking at the Moon. Using reasonable **"back-of-the-envelope"** reasoning, **estimate the distance from you to the Moon**. State which inputs you assume (e.g., the Moon's angular size and physical diameter) and explain your method. ```hint What relationship to use This is a **small-angle** problem. For a far object, angular size $\theta$ (in radians) relates physical diameter $D$ and distance $d$ by $\theta \approx D/d$. So which two inputs do you need to estimate? ``` ```hint The key anchor The whole estimate hinges on the Moon's **angular size** — roughly **half a degree** (the Sun is about the same, which is why solar eclipses just barely fit). Convert that to radians before dividing. You'll also need a rough physical diameter for the Moon. ``` ### 4) Stones in a boat A boat is floating in a lake with some stones sitting in it. You throw the stones overboard into the lake. Does the lake's water level **rise, fall, or stay the same**? Justify your answer. ```hint Compare displaced volumes Apply **Archimedes** to the two situations separately — stones in the floating boat vs. stones resting on the lakebed — and ask what volume of water each displaces. Whether the level rises, falls, or holds depends on which displaced volume is larger. What governs that comparison? ``` ### 5) Probability of an even product There are **two boxes**. Each box contains three cards labeled **1, 2, 3** (one of each). You draw one card from each box, **uniformly at random and independently**. What is the probability that the **product** of the two drawn numbers is **even**? ```hint Use the complement A product is **even** unless it's odd, and a product of integers is odd **only when every factor is odd**. So compute $P(\text{both draws odd})$ and subtract from 1. (You can verify by enumerating all $3 \times 3$ ordered pairs.) ``` ### 6) Uncompressed 1080p movies on a 2 TB drive Approximately how many **uncompressed 1080p** movies fit on a **2 TB** hard drive? Clearly state your assumptions, including: - **frame rate** (fps), - **color depth / bytes per pixel**, - **movie length**, and - the **definition of TB** (decimal vs. binary). ```hint Build the byte budget Multiply it out: $\text{pixels/frame} \times \text{bytes/pixel} \times \text{fps} \times \text{seconds}$ gives bytes per movie. Then divide the drive capacity by that. Resolution is $1920 \times 1080$; pick reasonable values for the other four assumptions. ``` ```hint Sanity-check the magnitude Uncompressed 1080p is **enormous** — on the order of tens of GB per minute — so don't be surprised if only a small handful fit. State how the count shifts if you change fps, bit depth, or decimal-vs-binary TB. ``` ### 7) Nested squares area Three squares are **nested** concentrically, each inner square formed by connecting the **midpoints** of the sides of the square around it. If the **innermost** square has area $A$, what is the area of the **outermost** square? ```hint The one fact you need Connecting the **midpoints** of a square's sides produces a new square. What's the ratio of the new square's area to the original? (Find the new side via the right triangle with legs $s/2$, $s/2$.) ``` ```hint Count the steps Each midpoint step multiplies area by a fixed factor. Going from outermost to innermost across **three** nested squares is **two** steps — so going *back* outward from area $A$ applies that factor's inverse twice. ``` --- ### Follow-up Questions After the main answers, expect the interviewer to probe how robustly you reason: - **Part 1:** How many tiles would you actually *order* once you account for cutting waste and breakage? How does the cut-tile count change if the tiles can't be cut at all? - **Part 3:** How sensitive is your Earth–Moon estimate to the angular-size assumption — if you were off by a factor of 2 on $\theta$, how far off is $d$? How would you measure the angular size more carefully with just your hand or a ruler? - **Part 5:** Generalize — if each box held cards $1$ through $n$, what's $P(\text{product even})$? What if you draw from $k$ boxes? - **Part 6:** Real codecs (H.264/H.265/AV1) compress video heavily. Roughly what compression ratio would make a typical 2-hour movie fit comfortably, and why is the uncompressed number so large? - **Part 7:** What's the area of the *n*-th nested square in terms of $A$? What does the side ratio (and rotation) converge toward? ### Constraints & Assumptions - The parts are **mutually independent** — solve each on its own; nothing carries over. - **Estimation parts (1, 3, 6)** are graded on stated assumptions and clean arithmetic, **not** a single "correct" number; an order-of-magnitude answer with sound reasoning is the goal. Reasonable defaults you may assume where not given: movie length ≈ 2 hours, frame rate 24–30 fps, 24-bit (3 bytes/pixel) color, and state whether "2 TB" means decimal ($2 \times 10^{12}$ B) or binary (TiB). - **Exact parts (2, 4, 5, 7)** have a single right answer — show the derivation, not just the figure. - Use consistent units throughout and call out conversions explicitly ($1\ \text{yd} = 3\ \text{ft}$; $1$ hour $= 3600$ s).