A Little Brainbone Exercise

November 2023
S	M	T	W	T	F	S
	1	2	3	4
5	6	7	8	9	10	11
12	13	14	15	16	17	18
19	20	21	22	23	24	25
26	27	28	29	30

OUR LINKS

YubaNet
White House Blog
Watts Up With That?
The Union
Sierra Thread
RL “Bob” Crabb
Barry Pruett Blog

A Little Brainbone Exercise

27 November 2023

George Rebane

Here are a couple of problems that our high school students taking the annual merit scholarship TechTest may encounter. The solutions require no higher math, actually only a little bit of lower math, and can be puzzled out on the back of an envelope. You, however, may use a full sheet of paper.

Problem 1: What are all the feasible clock times at which the minute and hour hands lie concurrently (on top of each other) on a standard clock face? Calculate your answers to the nearest second.

Problem 2: What are all the feasible clock times at which the counter-clockwise angle from the minute hand to the hour hand makes a right angle (90 degrees) on a standard clock face? Calculate your answers to the nearest second.

Hint: It’s possible (advisable?) to puzzle out the approach to both problems in your head before making a single mark on paper. But in any event, always draw a picture.

Posted in Critical Thinking & Numeracy

5 responses to “A Little Brainbone Exercise”

The Estonian Fox

30 November 2023

Problem 2:
In a 12-hour clock cycle, there will be (11) 90-degree angle times. So each new 90-deg arrives at (1 hour, + 3600sec/11) apart, or about 1hr, 5 minutes, 27.27 seconds between times. The minute hand has to travel a little bit farther each revolution to ‘catch up’ to the hour hand to make a 90-deg angle.
Times:
9:00:00
10:05:27
11:10:55
12:16:22
1:21:49
2:27:16
3:32:44
4:38:11
5:43:38
6:49:05
7:54:33
then back to 9:00:00
Do you know how long it took to cycle through rotating the stem on my Timex Easy Reader watch to do this? They should’ve made that stem about an inch in diameter.
Haven’t started #1 yet, but I suspect it will have the same 11 segments starting from 12:00:00.

LikeLike
The Estonian Fox

30 November 2023

Forgot to add:
I volunteer in 7th & 8th grade math/science classes. They don’t wear watches. And they don’t know what an analog watch is – a couple kids have complained that they wish there were audio announcements of the time, so they didn’t even have to look at the digital clock on the wall in the classroom. But when the bell rings – well, let’s say that Mr. Pavlov would have been proud of their reaction times.

LikeLike
George Rebane

30 November 2023

Efox – 427am – Very good. Your times tie with my spreadsheet calculated times to within one second. When you post the Prob#1 answers, could you please also describe a little about your approach – i.e. how you formulated the problem – since there are several ways to climb that hill.

LikeLike
The Estonian Fox

02 December 2023

Problem #1, and the general solution-
Starting position:
The minute hand at time t=0 is at a 90o angle in front of the hour hand. The minute hand will again be 90o in front of the hour hand when the angle is 450o (360o + 90o).
Minute hand travels 0.1o /sec = (12/120)o/sec; hour hand travels (1/120)o/sec
So when [((12/120)o/sec x t + 90o) – ((1/120)o/sec x t)] = 450o , we have the next occurrence of a right angle.
Solve for t:
t (12/120 – 1/120)o =360o,
Or t (11/120)o = 360o – and here is where the segments of 11 comes in.
t = 3927.273 seconds later for the next 90o angle, or 1 hour, 5 minutes, 27.273 seconds between occurrences.
This holds for ANY starting point of hands on the clock. Say you start at 4:00:00 (120o angle between the hands). 1 hour, 5 minutes, 27.273 seconds later the hands will again be 120o apart. Starting point = 120o , ending point is 480o = 360o of travel.
So for overlapping (coincident) hands starting at 12:00:00, the next times will be
1:05:27
2:10:55
3:16:22
4:21:49
5:27:16
6:32:44
7:38:11
8:43:38
9:49:05
10:54:33, then 12:00:00 again.

LikeLike
George Rebane

02 December 2023

Efox 1259pm – Correctamente! Interesting that you took a slightly different tack than I, but both approaches tie. I’ll post a pdf of my approach as an update.

LikeLike