# Oh, Never Mind

Zhuangzi and Huizi were strolling along the dam of the Hao Waterfall when Zhuangzi said, ‘See how the minnows come out and dart around where they please! That’s what fish really enjoy!’

Huizi said, ‘You’re not a fish — how do you know what fish enjoy?’

Zhuangzi said, ‘You’re not I, so how do you know I don’t know what fish enjoy?’

Huizi said, ‘I’m not you, so I certainly don’t know what you know. On the other hand, you’re certainly not a fish — so that still proves you don’t know what fish enjoy!’

Zhuangzi said, ‘Let’s go back to your original question, please. You asked me how I know what fish enjoy — so you already knew I knew it when you asked the question. I know it by standing here beside the Hao.’

Zhuangzi, China, fourth century B.C.

# More Geometry Trouble

Lewis Carroll offered this proof that all triangles are isosceles:

Let ABC be any triangle. Bisect BC at D, and from D draw DE at right angles to BC. Bisect the angle BAC.

(1) If the bisector does not meet DE, they are parallel. Therefore the bisector is at right angles to BC. Therefore AB = AC, i.e., ABC is isosceles.

(2) If the bisector meets DE, let them meet at F. Join FB, FC, and from F draw FG, FH, at right angles to AC, AB.

Then the triangles AFG, AFH are equal, because they have the side AF in common, and the angles FAG, AGF equal to the angles FAH, AHF. Therefore AH = AG, and FH = FG.

Again, the triangles BDF, CDF are equal, because BD = DC, DF is common, and the angles at D are equal. Therefore FB = FC.

Again, the triangles FHB, FGC are right-angled. Therefore the square on FB = the [sum of the] squares on FH, HB; and the square on FC = the [sum of the] squares on FG, GC. But FB = FC, and FH = FG. Therefore the square on HB = the square on GC. Therefore HB = GC. Also, AH has been proved equal to AG. Therefore AB = AC; i.e., ABC is isosceles.

Therefore the triangle ABC is always isosceles. Q.E.D.

2187 = (2 + 18)7

# The Clairvoyant Penny

Mathematician Thomas Storer offers a foolproof way to foretell the future: Flip a penny and ask it a yes-or-no question. Heads means yes, tails means no.

• Suppose the answer is yes. This is either true or false. If it’s true, then the original response was true. If it’s false, then the truth value of the original response is not false, i.e., it’s true.
• If the answer to the second question is no, this too is either true or false. If it’s true, then the original response was true. If it’s false, then the original response was not false, i.e., true.

Since all the outcomes agree, the penny’s original response is guaranteed to be correct.

# Logical

Raymond Smullyan doesn’t believe in astrology.

When asked why, he says, “I’m a Gemini.”

# A Cosmic Coincidence

Light travels 186,000 miles per second. The average diameter of Earth’s orbit is 186 million miles.

So, on average, sunlight reaches us in a neat 500 seconds.

# Digit Acrobatics

“Yields a falsehood when preceded by its quotation” yields a falsehood when preceded by its quotation.

# Easy as Pie

Now I will a rhyme construct,
By chosen words the young instruct.
Cunningly devised endeavour,
Con it and remember ever.
Widths in circle here you see,
Sketched out in strange obscurity.

Count the letters in each word.

# One Solution

The Professor brightened up again. ‘The Emperor started the thing,’ he said. ‘He wanted to make everybody in Outland twice as rich as he was before — just to make the new Government popular. Only there wasn’t nearly enough money in the Treasury to do it. So I suggested that he might do it by doubling the value of every coin and bank-note in Outland. It’s the simplest thing possible. I wonder nobody ever thought of it before! And you never saw such universal joy. The shops are full from morning to night. Everybody’s buying everything!’

— Lewis Carroll, Sylvie and Bruno