## The Angry Wife

Here is the homework problem I gave to my PRIMES STEP students.

Puzzle. A man called his wife from the office to say that he would be home at around eight o’clock. He got in at two minutes past eight. His wife was extremely angry at this lateness. Why?

The expected answer is that she thought he would be home at 8 in the evening, while he arrived at 8 in the morning. However, my students had more ideas.

For example, one student extended the time frame.

• The man was one year late.

Another student found the words “got in” ambiguous.

• He didn’t get into his house two minutes past eight. He got into his car.

A student realized that the puzzle never directly stated why she got angry.

• The wife already got angry when he said he would be home around eight, as she needed him home earlier.

The students found alternative meanings to “called his wife from the office” and “minutes.”

• He had an office wife whom he called. But the wife at home was a different wife, and she was angry.
• “Two minutes past eight” could be a latitude.
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Five elders are sitting around a large bonfire. They know that Oluf will put a hat of one of four colours (red, green, blue or yellow) on each elder’s head, and after a short time for silent reflection each elder will have to write down one of the four colours on a piece of paper. Each elder will only be able to see the colour of their two neighbours’ hats, not that of their own nor that of the remaining two elders’ hats, and they also cannot communicate after Oluf starts putting the hats on.
Show that the elders can devise a strategy ahead of time so that at most two elders will end up writing down the colour of their own hat.

2. #### Sanandan Swaminathan:

Here is a strategy that the elders can use. All 5 elders can use the same recipe below. The hat colors are B (Blue), G (Green), R (Red), and Y (Yellow). Assume that all 5 elders in the circle are facing the bonfire. An example of how to read the recipe is: If an elder sees a Green (G) hat to his right, and a Blue (B) hat to his left, then he should write Yellow (Y). This corresponds to the instruction GB -> Y below.

BB -> Y
BG -> B
BR -> B
BY -> Y
GB -> Y
GG -> B
GR -> B
GY -> Y
RB -> R
RG -> G
RR -> G
RY -> R
YB -> R
YG -> G
YR -> G
YY -> R

There are 4^5 = 1024 possible hat distributions. With every elder following the above recipe, 64 hat distributions lead to no elder writing down his own color, 768 hat distributions lead to exactly 1 elder writing down his own color, and the remaining 192 hat distributions lead to exactly 2 elders writing down their own colors.

I wrote a short program to make the machine discover a strategy, and it completed in a few seconds. There are 4 possible colors an elder can see to his right, and 4 possible colors he can see to his left, so there are 16 possibilities from each elder’s perspective. The leap of faith I took was that a strategy would actually exist where all 5 elders could use the same recipe. Another hypothesis was that the 16 states that an elder could see could be mapped to 4 states assigned to each of the 4 colors. If such a strategy indeed existed, the program would discover it. It turns out that the recipe shown above is not the only recipe the elders can use. Given below is the full set of 12 recipes found by the program. For brevity, I have pasted each recipe as a list. The first line shows the list of 16 possible states that an elder can see. For any given state he sees (combination of the hat color he sees to his left and the hat color he sees to his right), the elder should write down the color at the same index in the recipe.

[‘BB’, ‘BG’, ‘BR’, ‘BY’, ‘GB’, ‘GG’, ‘GR’, ‘GY’, ‘RB’, ‘RG’, ‘RR’, ‘RY’, ‘YB’, ‘YG’, ‘YR’, ‘YY’]

Recipes:

[‘Y’, ‘B’, ‘B’, ‘Y’, ‘Y’, ‘B’, ‘B’, ‘Y’, ‘R’, ‘G’, ‘G’, ‘R’, ‘R’, ‘G’, ‘G’, ‘R’]
[‘Y’, ‘B’, ‘B’, ‘Y’, ‘G’, ‘R’, ‘R’, ‘G’, ‘Y’, ‘B’, ‘B’, ‘Y’, ‘G’, ‘R’, ‘R’, ‘G’]
[‘R’, ‘B’, ‘R’, ‘B’, ‘R’, ‘B’, ‘R’, ‘B’, ‘Y’, ‘G’, ‘Y’, ‘G’, ‘Y’, ‘G’, ‘Y’, ‘G’]
[‘R’, ‘B’, ‘R’, ‘B’, ‘G’, ‘Y’, ‘G’, ‘Y’, ‘G’, ‘Y’, ‘G’, ‘Y’, ‘R’, ‘B’, ‘R’, ‘B’]
[‘G’, ‘G’, ‘B’, ‘B’, ‘Y’, ‘Y’, ‘R’, ‘R’, ‘G’, ‘G’, ‘B’, ‘B’, ‘Y’, ‘Y’, ‘R’, ‘R’]
[‘G’, ‘G’, ‘B’, ‘B’, ‘R’, ‘R’, ‘Y’, ‘Y’, ‘R’, ‘R’, ‘Y’, ‘Y’, ‘G’, ‘G’, ‘B’, ‘B’]
[‘Y’, ‘Y’, ‘R’, ‘R’, ‘B’, ‘B’, ‘G’, ‘G’, ‘B’, ‘B’, ‘G’, ‘G’, ‘Y’, ‘Y’, ‘R’, ‘R’]
[‘R’, ‘R’, ‘Y’, ‘Y’, ‘B’, ‘B’, ‘G’, ‘G’, ‘R’, ‘R’, ‘Y’, ‘Y’, ‘B’, ‘B’, ‘G’, ‘G’]
[‘Y’, ‘G’, ‘Y’, ‘G’, ‘B’, ‘R’, ‘B’, ‘R’, ‘B’, ‘R’, ‘B’, ‘R’, ‘Y’, ‘G’, ‘Y’, ‘G’]
[‘R’, ‘G’, ‘G’, ‘R’, ‘B’, ‘Y’, ‘Y’, ‘B’, ‘R’, ‘G’, ‘G’, ‘R’, ‘B’, ‘Y’, ‘Y’, ‘B’]
[‘G’, ‘Y’, ‘G’, ‘Y’, ‘G’, ‘Y’, ‘G’, ‘Y’, ‘B’, ‘R’, ‘B’, ‘R’, ‘B’, ‘R’, ‘B’, ‘R’]
[‘G’, ‘R’, ‘R’, ‘G’, ‘G’, ‘R’, ‘R’, ‘G’, ‘B’, ‘Y’, ‘Y’, ‘B’, ‘B’, ‘Y’, ‘Y’, ‘B’]

3. #### Andew:

Great article. I’ve been spending more and more time on the blog lately