Brain Teaser Case: You look at a clock and notice that it is 3:15 in the afternoon. How many degrees apart are the hour and the minute hand?


Solution: At 3:15, the minute hand is exactly at the quarter point of the clock face, dead east and ninety degrees to the right of noon. However, the hour hand has moved ever so slightly. The key question is, how much has it moved? Here's one approach...

  • There are 360 degrees in a circle
  • There are 12 hours on a clock face
  • Therefore each hour is equivalent to 30 degrees
  • In a quarter of an hour, there are ?of those 30 degrees (30 divided by 4 = 7 ?
  • Therefore, at 3:15, the hands are 7 ?degrees apart


Market Sizeing Case: How many taxis are there in New York City?


Approach: Think through the information you do have. Having spent a weekend in New York once, you may remember that there are five boroughs. You may have also noticed, however, that the vast majority of taxis are in Manhattan, with a smaller number at LaGuardia and JFK airports.


Considering Manhattan, then ... conveniently, most of Manhattan has a grid-like street structure. The avenues are the north-south streets, which tend to be quite wide, typically 4-5 lanes. The numbered streets, running east-west, are more narrow and often only have 1-2 lanes of traffic. During rush hour, when most taxis are presumably out, roughly how many taxis are on a given block of an avenue or a street?


On the avenues, there are probably 3 taxis per lane, for a total of 12-15 taxis per block of avenue. On the streets, there are probably 4 taxis per lane (blocks are rectangular, with the street side longer than the avenue side), for a total of 5-8 per block.

How many block of streets and avenues are there then? Estimate that there are roughly 15 avenues (any large street running north-south). There are approximately 150 east-west streets in the main part of Manhattan. So that means there are 15 x 150 = 2,250 blocks worth of avenues and streets.


But don't forget that Central park covers a large area, roughly 50 blocks long and 3 blocks wide. So we need to reduce our blocks estimate by the 3 x 50 = 150 blocks in the park. 2,250-150 = 2,100 city blocks. To make the math easier, round to 2,000. For the avenues, we determined there were 12-15 taxis per block. So 15 x 2,000 = 30,000 taxis on the avenues at rush hour, and 8 x 2,000 = 16,000 taxis on streets. 30,000 + 16,000 = 46,000 taxis in Manhattan.


One last detail ?what about bridges, tunnels and airports? Estimate that there are 100 taxis somewhere on the premises of each of LaGuardia and JFK. That adds 200 to the number. Also assume that on each of the main bridges and tunnels which connect Manhattan to the airports, there are 100 taxis on average. If there are roughly 3 bridges or tunnels which are considered routes to the airports, that makes 300 additional taxis.


Adding an additional 500 for the taxis on the freeways between tunnels/bridges and the airport, waiting in line at the train station, etc., our total number comes to 46,000 + 200 + 300 + 500 = 47,000.