2  Mathematical problems.
   In this session the mcp-abacus should be used to do all the calculations.
   Show me what calls you made to answer this question.

  I'm boxing up 230 home-made cookies into bags of 7 to sell at the fair. How
  many full bags can I make, and how many cookies will be left over for me to
  nibble? 
  
  A school outing has 154 children and each minibus seats 17. How many minibuses
  do we need to book so that nobody gets left behind? 
  
  Dinner came to 47 euros and I want to leave an 18% tip, but I'd like to hand
  over a round number of euros to keep the cash simple. What should I pay,
  rounded to the nearest euro?
  
  My fuel receipt says I pumped 38.6 litres, but my mileage logbook only takes
  whole litres. What do I write down?

  I left a bottle of fizzy water open and it goes flat steadily, losing about
  the same share of its bubbles every hour. If it lost a quarter of its fizz in
  the first hour, how much is still left after a six-hour evening? 

  My savings sit in an account that grows continuously at a steady yearly
  rate of 4%. If I leave 5,000 in it untouched for three years, what will it
  have grown to? 

  My back garden drops by about 1 metre over a 12-metre run from the house to
  the fence. How many degrees of slope is that? 

  A delivery ramp rises 30 centimetres, and the sloped surface of the ramp
  itself is 2 metres long. What angle does it make with the ground, in degrees?

  My last six electricity bills were 88, 92, 79, 105, 96 and 84 euros. How
  much do my bills typically swing from one month to the next? 

  I timed my morning jog on five different days and got 27, 31, 26, 29 and
  33 minutes. How consistent am I — what's the typical spread in those times? 

  Walking to the station I go 600 metres along one street, then turn the
  corner and go 800 metres along the next.  If there were a footpath cutting
  straight across, how long would that shortcut be?

  I want to check whether a new monitor will sit on my desk. The desk is 70
  centimetres deep and 140 centimetres wide — what's the straight-line distance
  from one corner to the opposite one? 

  A rumour at my office doubles the number of people who've heard it every
  single day, starting from just me. How many days until the whole 1,000-person
  building knows? 

  My friend texts that she's 200 metres east and 150 metres south of where I'm
  standing in the park. What compass bearing do I turn to so I'm facing straight
  at her? 

  I've got 48 red and 36 blue balloons and want to make identical party bundles
  using every balloon, with as many bundles as possible. How many bundles, and
  what's in each? 

  Two buses pull out of the depot together; one returns every 12 minutes, the
  other every 18. How long until they next leave at the very same moment? 

  I have six framed photos and one long shelf. How many different left-to-right
  orders could I arrange them in?

  Eight sprinters line up for the final. In how many different ways can the
  gold, silver and bronze medals end up being handed out?

  From my ten-person book club we draw four names to host next month's meeting.
  How many different groups of four could we end up with? 

  Where a value sits — percentile

  The ten kids in the class scored 52, 61, 68, 70, 73, 75, 80, 84, 88 and 95 on
  the test. What score marks the cut-off for the top tenth of the class? 

  My investment grew 10% the first year, dropped 20% the second, then grew 30%
  the third. What single steady yearly growth rate would have left me exactly
  where I ended up?

  I drove out to the coast at 60 km/h and crawled back along the very same road
  at 40 km/h. What was my average speed for the round trip? 

  Over several weeks I jotted down each day's temperature and how many ice
  creams my van sold. Do hotter days really go hand-in-hand with more sales, and
  how tightly? 

  I'm borrowing 15,000 for a car at 6% a year, paying it off in equal monthly
  instalments over four years. How much is each monthly payment? 

  I put €1,000 in a savings account earning 5% a year, compounded annually.
  How many years until it doubles to €2,000?

  My candle business has €450 of fixed monthly costs and each candle costs €3
  to make but sells for €12. How many candles must I sell each month to break
  even?

  I have 40 metres of fencing to make a rectangular vegetable patch against my
  house wall (only three sides need fencing). What width gives me the biggest
  area?

  I want a cylindrical tin can to hold exactly 500 ml. What radius uses the
  least metal?

  A closed cylindrical drum must hold 100 litres. What radius and height
  minimize the metal used?

  I walk east at 5 km/h from a point; my friend walks north at 4 km/h from 10
  km east of me. When are we closest, and how close?

  My orchard yields about 50*t - 5*t**2 crates if I wait t weeks (fruit grows
  then rots). When is the yield highest?

  I have €1,000 for social and search ads. Reach is roughly 80*sqrt(s) +
  60*sqrt(g), but I want to penalize blowing the budget. How should I split the
  spend?

  One friend is at (2, 9) and another at (10, 1) on the map. We want a café
  that minimizes the longer of our two travel distances squared, roughly the
  sum of squared distances. Where?

  A Uniswap V2 pool holds 1,500.000000000000000000 ETH and
  4,200,000.000000000000000000 USDC. With the 0.3% swap fee, exactly how much
  ETH must I sell to receive 100,000.000000000000000000 USDC out — to the full
  18 decimals?

  What's 0.1 + 0.2 in IEEE-754 double — show me the rounding error.

  What's sqrt(2) * sqrt(2) in floating-point vs rational?

  Solve x**2 - 2 = 0 for x between 1 and 2.

  Find x and y that minimize (x-1)**2 + (y+2)**2, x in [0,5], y in [-4,2].

  At what price does demand 100 - 2*p equal supply 20 + 3*p?

  What interest rate makes a €1,000 loan repaid as €1,200 after 4 years?

  What production level q minimizes my average cost q**2 - 10*q + 40?

  A 2 m pole casts a shadow. At what sun elevation angle is the shadow exactly
  3 m long?

  My 8 m rafter needs the roof peak to be 2.5 m above the wall plate. What
  pitch angle does that require?

  My coffee comes out at about ninety degrees and the kitchen sits at twenty.
  It cools fast at first and then slows down, the usual way hot drinks do.
  After 5 minutes it is 82.1 degrees.  Roughly how many minutes before it's
  down to a drinkable sixty degrees?

  What's the average and standard deviation of 4, 8, 15, 16, 23, 42?

  I've leaned my six-metre ladder against the wall and the feet are about a
  metre and a half out from the bottom. Roughly what angle is it making with
  the ground? People say steeper than seventy-five degrees is dodgy.

  What's π to 100 decimal places?

  Give me the value of ln(10) out to 85 digits.

  You borrow 10,000 dollars and pay it back in 24 equal monthly installments of
  470 dollars each. What constant monthly interest rate makes the loan balance
  reach exactly zero on the last payment?  
    
  A comet travels an elliptical orbit with eccentricity 0.7. Exactly one quarter
  of the way through its orbital period after passing closest to the sun, what
  is its eccentric anomaly in radians?

  Find the positive number that, when raised to the fifth power, equals seven
  times the number plus three.

  You cut four equal squares out of the corners of a 20 by 30 centimeter sheet
  of cardboard and fold the flaps up into an open box. What side length for the
  cut-out squares gives the box a volume of exactly 1000 cubic centimeters? Give
  the smallest such length.
 
  A lifeguard is standing on a straight beach. A swimmer is 40 meters straight
  out from a point that lies 50 meters down the beach from the lifeguard. The
  lifeguard runs along the sand at 6 meters per second and swims at 1.5 meters
  per second. How far down the beach should the lifeguard run before entering
  the water to reach the swimmer in the least time?

  A power cable 25 meters long hangs freely between two towers that are 20
  meters apart and at the same height. How far below the attachment points does
  the lowest point of the cable hang?

  A hallway 2 meters wide meets another hallway 1.5 meters wide at a right
  angle. What is the length of the longest thin pole that can be carried flat
  around the corner?

  A drilling platform sits 6 kilometers off a straight coastline.  The receiving
  station is onshore, 12 kilometers along the coast from the nearest point on
  land.  Underwater pipe costs 5 million dollars per kilometer and land pipe
  costs 3 million. At what point on the shore should the pipe land to make the
  project cheapest?


  Tier 1 — things I had to work around this session

  - Sample stddev / variance (or a sample=true variant) — stddev is
    population-only, so for the 4·8·15·16·23·42 set I had to multiply by
    sqrt(n/(n−1)) by hand to get the n−1 version. Both conventions are needed
    often enough to be first-class.
  - hypot(x, …) (Euclidean norm) — the ladder, rafter, and café-distance
    problems all reduce to √(Σ squares). A hypot avoids spelling out sqrt(a**2 +
    b**2) and is numerically safer.

  Tier 3 — solver / analysis upgrades (bigger lifts, high leverage)

  - Native solver constraints (equality + inequality) — the €1,000 ad-budget
    problem forced me to hand-roll a −50*max(0, s+g−1000)² penalty. A real
    subject_to would be cleaner and less hacky than tuning penalty weights.

  Tier 4 — domain packs (optional, fits your usage)

  - Finance: npv, irr, fv, pv, pmt — the loan/compound-interest and (implicitly)
    the ad-ROI problems are exactly these. Given how much finance/crypto flows
    through this tool, a small finance pack would punch above its weight.


