Re: OT: A puzzle -

Quentin Grady wrote:
Think of a number between 7 and 100.

Divide by five. What is the remainder? Two I answered. Divide by four What is the remainder? One I answered. Divide by seven. What is the remainder? Zero.

The number is 77, my friend replied with little hesitation.

How much hesitation? It took me fifteen minutes
to come up with a method, but the first time
I used it, I had less than a second of hesitation:

(a) X mod 5 = 2 : last digit must be 7 or 2

(b) X mod 4 = 1 : X must be odd ( _7 )

(c) X mod 7 = 0 : X must be 7 or 77

(d) 7 mod 4 = 3 : X must be 77

For (c) you only need to know the sevens--if you had said
remainder six, I would know that X + 1 must be 28 or 98.
If 0 < X < 100, then (c) has at most two choices, and
since the difference between them is seventy, only one
can have the desired mod four value.

How did I do all that in less than a second?
I didn't! My son picked a number, and I said,
"Divide by five and tell me the remainder."

He also said "Two," and so I knew the last digit
was either two or seven _before_ I finished saying,
"Divide by four and tell me the remainder."

He said "Two," and so I knew the number was
even (last digit two) _before_ I finished saying,
"Divide by seven and tell me the remainder."

He said, "Four," so in the time it takes to say,
"Your number is," I figured that it had to be
three less than a multiple of seven that ends in
five. There's only one choice: 32, so I did not need
the tie-breaker (d).

But I was wrong! The number is 92! How did that happen?

The instructions are not clear. His computations were:

Divide by five and tell me the remainder:
92 / 5 = 18 R 2

Divide by four and tell me the remainder:
18 / 4 = 4 R 2

Divide by seven and tell me the remainder:
4 / 7 = 0 R 4

which my method turns into
(a) X mod 5 = 2 : last digit must be 7 or 2
(b) X mod 4 = 2 : X must be even ( _2 )
(c) X mod 7 = 4 : X+3 can only be 35
: X = 32

If I had bothered to check
(d) 32 mod 4 = 0
then I might have realized that he did not do what I wanted.

Instead, I said, "Your number is 32" and he said "92"
I said "92 divided by seven has a reminder of one."
He said, "Oh, I thought you meant....."

Before I even worked out the technique, I had realized two
possible interpretations of the instructions, so if I had
checked step (d), I would have suspected what happened, and
I could have taken a little more time to figure:

Third remainder is also the dividend (divide X < 100 by 5,
second dividend < 20, divide that by four: less than
five).

So second dividend is 4 * 4 + 2 = 18 and
the number is 18 * 5 + 2 = 92

However, with a little more practice and thought, I would
have rephrased the instructions to make the alternate
method unnecessary.

--
Wes Groleau
Heroes, Heritage, and History
http://UniGen.us/PGV
.

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