Thursday, 14 December 2017

On The 'Mystery Calculator' Trick

'The Mystery Calculator' trick is a perennial, favourite 'surprise gift' found in Christmas crackers[1]  — along with the (hard) plastic moustache/slug, the fortune-telling fish, and perhaps (if you're lucky) a die.

For a maths teacher, 'The Mystery Calculator' is a potent conceit that piques students' interest and that — when used carefully[2] of course — levers and encourages strong, mathematically mature thinking.  After playing the trick out on/with students, it is demystified carefully, and collectively, before students use their newfound knowledge and understanding creatively in devising a different trick, based on the same principles.  (A ppt version of the cards, one card per slide, can be downloaded here.)



The Trick:
  • Ask someone to choose a number from any of the six cards.
  • Show them each card in turn and ask them if their number appears on it.
  • Add the numbers in the top left-hand corner of each card that contains their number.
  • The total is their number.
















    The How / Why:

    The number 1 in binary is (1 x 2⁰)  → (1)10 = (1)2  → Hence the number 1 appears on the 1st card only.

    The number 2 in binary is 10... (1 x 2¹) + (0 x 2⁰)  → (2)10 = (10)2  → Hence the number 2 appears on the 2nd card, but does not appear on the 1st or any other.

    The number 3 in binary is 11... (1 x 2¹) + (1 x 2⁰)  → (3)10 = (11)2  → Hence the number 3 appears on the first 2 cards, but does not appear on others.

    The number 4 in binary is 100... (1 x 2²) + (0 x 2¹) + (0 x 2⁰)  → (4)10 = (100)2  → Hence the number 4 appears on the 3rd card but does not appear on the 1st or 2nd, or any other.

    The number 5 in binary is 101... (1 x 2²) + (0 x 2¹) + (1 x 2⁰)  → (5)10 = (101)2  → Hence the number 5 appears on the 1st and 3rd cards, but does not appear on the 2nd, or any other.

    The number 6 in binary is 110... (1 x 2²) + (1 x 2¹) + (0 x 2⁰)  → (6)10 = (110)2  → Hence the number 6 appears on 2nd and 3rd cards but does not appear on the 1st, or any other.

    The number 7 in binary is 111... (1 x 2²) + (1 x 2¹) + (1 x 2⁰)  → (7)10 = (111)2  → Hence the number 7 appears on on 1st, 2nd and 3rd cards, but does not appear on any other.



    The number 41 in binary is 101001... (1 x 2⁵) + (0 x 2⁴) + (1 x 2³) + (0 x 2²) + (0 x 2¹) + (1 x 2⁰)  → (41)10 = (101001)2  → Hence the number 41 appears on the 1st, 4th and 6th cards but does not appear on the 2nd, 3rd or 5th.



    The number 62 in binary is 111110... (1 x 2⁵) + (1 x 2⁴) + (1 x 2³) + (1 x 2²) + (1 x 2¹) + (0 x 2⁰)  → (62)10 = (111110)2  → Hence the number 62 appears on the 1st, 2nd, 3rd, 4th and 5th cards, but does not appear on the 6th.

    The number 63 in binary is 111111... (1 x 2⁵) + (1 x 2⁴) + (1 x 2³) + (1 x 2²) + (1 x 2¹) + (1 x 2⁰)  → (63)10 = (111111)2  → Hence the number 63 appears on all six cards.



    Thinking through some possible starting questions for students to consider (or to consider with students), before moving on perhaps with students creating their own 'Mystery Calculator' trick (see also [3]):
    • What do you notice about the numbers on each card?
    • Do these patterns matter — what could we do to see if they do?
    • Can we have a number larger than 63 on any of the cards?
    • What about all of the other numbers on the cards — can they be chosen randomly?
    • What number would appear on all cards if there were four, five, seven, ten, ... cards?  
    • Does the number in the top left-hand corner have to be there?
    • Are there any constraints to the numbers we place on the cards?
    • Would a similar trick in another base, maybe base-3, be more magical?
    • How high can we count in binary on our fingers? (Watch this Ted-Ed video from James Tanton.)





    Notes & (Select) Links:

    [1]  See here for an example, or here.

    [2]  See 'When Magic Fails in Mathematics,' by Junaid Mubeen.

    [3]  See this by Katie Steckles in The Aperiodical'On Disreputable numbers'.

    For other Christmathsy problems to consider with students, try the 'Santamaths' problem, the '12 Days of Christmas' problem, see this selection from @mathsjem, and have a look at my 'xmaths card'.


    Tuesday, 5 December 2017

    On Mathematics To Listen To



    A collection of links to radio and podcast episodes (in English) with a mathematics bent — to inspire teachers to inspire, to feed the minds of our young mathematicians in the making, or, simply, to entertain.  Click here or on the 'Audio' tab above for the complete list.  Programmes have been (loosely) categorised according to themes below, and arranged alphabetically within each category by programme title.

    The collection includes, in no particular order:
    1. Select episodes from BBC Radio 4 In Our Time 
    2. All episodes from BBC Radio 4 Marcus du Sautoy's A Brief History of mathematics
    3. All episodes from BBC Radio 4 Marcus du Sautoy's Five Shapes
    4. All episodes from BBC Radio 4 Simon Singh's Five Numbers
    5. All episodes from BBC Radio 4 Simon Singh's Another Five Numbers
    6. All episodes from BBC Radio 4 Simon Singh's A Further Five Numbers
    7. Select episodes from BBC Radio 4 The Infinite Monkey Cage
    8. All episodes from BBC Radio 4 A History of the Infinite
    9. Select episodes from BBC Radio 4 More Or Less
    10. Select episodes from BBC World Service More Or Less
    11. Select episodes from BBC Radio 4 Seriously
    12. Select episodes from BBC Radio 4 Incarnations: India in Fifty Lives
    13. Select episodes from BBC Radio 4 Start the Week
    14. Select episodes from BBC Radio 4 The Life Scientific
    15. Other programs from BBC Radio
    16. Select episodes from The Partially Examined Life podcast
    17. Select episodes from Futility Closet podcast
    18. Select episodes from Freakonomics podcast
    19. Select episodes from Radiolab podcast
    20. Select episodes from Teach Better podcast
    21. Select episodes from Modern Learners podcast
    22. Select episodes from Plus Magazine podcasts
    23. Select episodes from The Guardian's Science Weekly podcast
    24. Select episodes from The Naked Scientists podcast
    25. Select episodes from The NCETM Maths podcast
    26. Select episodes from The James Altucher Show podcast

    There are, of course, many other superb podcasts from which episodes have not been included in this selection, purely because as their sole focus is on mathematics or its teaching, I would be listing links to all of their episodes.  (Episodes in this selection are from series that don’t concentrate solely on mathematics, or from shorter series that concentrate on one aspect of mathematics.)  

    I have listed a selection of some other, excellent podcasts solely focused on mathematics and/or its teaching here, but of course this list is not exhaustive.  If you are aware of other podcasts or audio to point people in the direction of, do let me know.