Quod Erat Demonstrandum

2009/05/13

數數唸

一.

以下數字是「旋轉對稱」的嗎?

1961

是?嗯,問題係「旋轉對稱」應該是描述圖形,而不是數字…如果我把它寫成

1961

那這個數字還是「旋轉對稱」的嗎?

二.

f(x) is differentiable at x = a”does NOT imply ”f'(x) is continuous at x = a

Just think about a classic example

f(x) = x^2\sin \frac{1}{x} for x \ne 0 and f(0) = 0.

三.

Justin had just sent me an interesting question.

Prove by mathematical induction that x^3 + y^3 + z^3 = 3^n has integral solution (x,y,z) for any positive integer n.

Since

1^3 + 1^3 + 1^3 = 3^1
0^3 + 1^3 + 2^3 = 3^2
0^3 + 0^3 + 3^3 = 3^3

the statement is true for n = 1,2,3.

For any positive integer k,
suppose x_0^3 + y_0^3 + z_0^3 = 3^k for some integers x_0, y_0, z_0,
then (3x_0)^3 + (3y_0)^3 + (3z_0)^3 = 3^{k+3}
i.e. the statement is true for n = k + 3

Another problem is, whether, for fixed n, x^3 + y^3 + z^3 = 3^n has unique integral solution (up to permutation)? Could you give a proof or counter-example? Thank you in advance.

四.

在早上回校的小巴,從坐在最後一個座位的一刻開始,我一直被迫地,聽司機和某位阿叔的對話:

司:「如果買返上一期果幾個冧巴(number),今期(六合彩)就可以中三個字啦!」
叔:「所以話,你要『枕住』幾期買同一注冧巴囉,起碼連續五期要咁買!」
司:「你會唔會買電腦飛?」
叔:「唔好買電腦飛呀!電腦飛每次都有幾個冧巴唔出o架!」
司:「唔係掛?」
叔:「真o架,我個 friend o係馬會做,佢話o架!即係如果佢唔出 3,6,9,今期開 3,6,9,咁咪xx」

期間,小巴鍾聲響過,飛站後,小姐說:「有落呀,頭先o禁o左鐘o架!」

停車,未幾。

司:「我聽唔到有人打鐘。」
叔:「我都聽唔到!」

(\ ___ /)

鐘(擬人法):「我肯定頭先畀人打過,阿 John 可以做證!」

為免他們因吹水而再度飛站,我唔打鐘,高聲揚:「橋底有落!」司機舉手示意。

將要到站,司機問:「有冇人要落?」

我(\ ___ /):「有!」

[SBA]

1.「連續 N 期買同一注」是個買「六合彩」的好策略嗎?
2.「電腦彩票每次都有幾個數字不出現」如何影響中獎的機會?
3. 試比較評論以下的命題:
(a)「連續買 19 次六合彩都唔中頭獎,咁買第 20 次都唔中頭獎的機會便很高!」
(b)「擲一元硬幣 19 次,每次結果都是『公』,咁擲第 20 次都是『公』的機會便很高!」

1 則迴響 »

  1. Another problem is, whether, for fixed n, x^3 + y^3 + z^3 = 3^n has unique integral solution (up to permutation)? Could you give a proof or counter-example? Thank you in advance.

    For n = 1, we have (x, y, z) = (1, 1, 1), and (4, 4, -5). It is an open problem whether these are all solutions.

    迴響 由 koopakoo — 2009/05/16 @ 1:04 下午 | 回覆


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