Quod Erat Demonstrandum

2008/04/30

今天又代堂:神奇數字,魔術,Gag

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 4:00 下午

進入 F.3D 班房

1. 顯示聖經約翰福音(若望福音)第廿一章的內容,指出神奇的數字 153。介紹當中的特性(全班幫我『督機』驗算)。(參考 Sarah 在 mathdb 的 post

2. 引入另一個神奇數字:6174
給定任意一個 4 位數(數碼不盡相同者,即不容 9999 之類),把 4 個位數重組,得最大及最小的數(比如 1985,最大者為 9851,最小者為 1589),找出最大最小數之相差(即 9851 – 1589),不斷進行如此的運算。例如
9851 – 1589 = 8262
8622 – 2268 = 6354
6543 – 3456 = 3087
8730 – 0378 = 8352
8532 – 2358 = 6174
得到 6174。但原來,無論開始是用哪個 4 位數,最終,一定得到 6174。(不過,若不幸,要用較多的步驟才得結果的話,同學會沒有耐性試。)

3. 介紹至今仍是謎一樣的:3x+1 猜想
給定任意一個正整數,若它是雙數,除 2;若它是奇數,把它乘 3 後加 1(即 3x+1 之)。重複這個算法。舉例
給定 22,我們有
22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
無論給定的正整數是什麼,最終的答案『似乎』都是 1。(說『似乎』,是因為至今仍未找到證明或反例)

4. 高潮位:教他們下面的樸克魔術『蓋綿被』(見下)

5. 見有時間,匆匆談 65 = 64 的證明及『隨意』說一些 gag,包括

及在濟濟一堂的舊貼圖:
http://www.hkedcity.net/ihouse_tools/forum/read.phtml?forum_id=27877¤t_page=&i=960290&t=957675&v=t

就這樣,彼此在毫無壓力下,上完這堂『數學課』。

8 則迴響 »

  1. 你比我地個website 我同sister 岩岩read 晒, 佢話你都好好笑… anyway, I just feel like that you like gags, but you avoid telling one in class.
    By the way I’m having a talent show of piano on Friday after off, dunno if you have time. Oh but if you are just a bit busy, I HIGHLY RECOMMEND you not to come coz the piano they gave us is an electronic one, which has the worst dynamics (大細声), ever.
    Night, my eyelids aren’t responding

    迴響 由 Edmund — 2008/04/30 @ 10:51 下午 | 回覆

  2. Edmund,

    Hope you’ll have a good show on Friday. I try to go there if I have time. Just wanna know what is the purpose of “天才表演" organized by SU. Is that something for the Talentine next year? Or just to make a little bit fun of it?

    For gags, I believe that most of people like them.

    迴響 由 johnmayhk — 2008/05/01 @ 5:45 下午 | 回覆

  3. Well, it’s just fun

    迴響 由 Edmund To — 2008/05/01 @ 5:52 下午 | 回覆

  4. 2. 引入另一個神奇數字:6174

    I discovered that [0246] [1357] [2468] [3579]
    these few numbers,

    找出最大最小數之相差 :
    only one time,
    we can find out 6174

    [0246] : 6420-0246 = 6174
    [1357] : 7531-1357 = 6174
    [2468] : 8642-2468 = 6174
    [3579] : 9753-3579 = 6174

    迴響 由 yU — 2008/05/01 @ 7:55 下午 | 回覆

  5. Good observation, YU.

    經你說一說,我嘗試推論一下,不難發現:要『一步到位』地得到 6174,我們試的四位數,只要它的最大和最小數碼相差 6,餘下的兩個數碼相差 2,便可。即是以下的四位數,都是『一步到位』地得到 6174 的,試試看。

    0026
    0136
    0246
    0356
    1137
    1247
    1357
    1467
    2248
    2358
    2468
    2578
    3359
    3469
    3579
    3689

    這樣,我們可以繼續研究:

    (a)什麼四位數可以『兩步到位』?
    (b)什麼四位數可以『N步到位』?(N 的最大值是多少)
    (c)如何證明這個所謂『6174 現象』?
    (d)除了6174,還有沒有其他數字存在類似現象?

    迴響 由 johnmayhk — 2008/05/01 @ 10:57 下午 | 回覆

  6. for question (a):
    I only found
    8820-0288=8532
    8532-2358=6174

    for question (b):
    I still need more time to think about it

    for question (c):
    I don’t know why,
    but I discovered that 6174:
    7641-1467=6174

    for question (d):
    I found the number “495″
    is also like “6174″,
    but in 3-digits numbers.

    I also discovered when it is 5 digits,
    the middle number always be “9″
    and the calculated number keep on repeating

    eg.[1]
    95521-12559=82962
    98622-22689=75933
    97533-33579=63954
    96543-34569=61974
    97641-14679=82962
    98622-22689=75933

    eg.[2]
    98765-56789=41976
    97641-14679=82962
    98622-22689=75933
    97533-33579=63954
    96543-34569=61974
    97641-14679=82962
    98622-22689=75933

    eg.[3]
    96642-24669=71973
    97731-13779=83952
    98532-23589=74943
    97443-34479=62964
    96642-24669=71973
    97731-13779=83952

    The result:
    the 5-digits numbers,
    I found that the calculated value is keep on repeating.
    (4 different numbers a cycle)

    迴響 由 yU — 2008/05/02 @ 12:44 上午 | 回覆

  7. Marvelous, yU! Keep on, you may work out something new!

    迴響 由 johnmayhk — 2008/05/02 @ 8:11 上午 | 回覆


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