Heaths Dice

FROM https://islandsofmath.wordpress.com/2016/10/05/di-ciphering-dice-2/

FROM https://islandsofmath.wordpress.com/2016/10/05/di-ciphering-dice-2/

## Di-Ciphering Dice
Posted in Magic, Probability, TeachingIn 1927 Royal V. Heath marketed a magic effect called “The
Di-Ciphering Trick” based on a math trick developed by Ed Balducci
(1906-1988); a New York City magician and civil engineer by day. Royal
Vale Heath (1883-1960) was a New York City broker, American Magician and
puzzle enthusiast. Tricks with serial numbers, Magic Squares and dice
were his specialty. The “Di-Ciphering” trick consists of 5 dice cubes bearing a different 3 digit number on each face – 30 numbers in all. A spectator would roll the dice and the magician would quickly announce the sum of all the numbers. Let’s look at the dice used for the trick: An example roll of the dice is: The method used to quickly calculate the sum of the 5 die of 3 digit numbers is: This will work with any roll of the dice. Why does this method work? Let’s look at some of the 7,776 sums (a small sample) and look at the numbers on the dice as well to see if we can find any patterns.
Observations: - Sum of middle digit of the numbers (one from each dice) always add to n0 – this needs to occur for the last two digits of sum of the numbers to be sum of last digits.
- Middle digit of:
D1 = 8, D2 = 4, D3 = 5, D4 = 6, D5 = 7 thus 8 + 4 + 5 + 6 + 7 = 30 - Therefore sum of last digits will be last two digits of the sum of numbers from each of the 5 dice.
- All combinations of sums of 5 dice need to have first 2 digits plus last 2 digits always add to 50.
- Sum of first and last digit for each dice:
D1 = 7, D2 = 8, D3 = 13, D4 = 9, D5 = 10 This sum is 47 - Sum of first digit plus 3 (since middle digits always add to 30) + sum of last digits for each combination of dice = 50.
- So we can determine sum of all the dice on the roll by adding last digits for the left 2 digits of the sum then subtract from 50 to get the leading two digits of the sum.
- Sum of digits of the total is always 14
We know there can be 7,776 possible totals when we roll the dice.
A different trick using these dice was given in the book Practical Mental Magic by Theodore Annemann.
The Trick: Preparation and Routine: The use of the dice make the test appear very fair. There is never a thought that in the moment of putting the dice in line, or in instructing the spectator what to do, you have learned the total by the short cut process possible with this trick. The opinion they have is that there can be hundreds of variations. If there is a question about the digits on the dice you can mention that they are used for some money game (without going into that part further) as an excuse for their being numbered with three digits to a side. As further preparation for the trick we need to look at all of the
unique somes in a slightly different manner. We are interested in the
first two digits and last two digits of the sums as pairs with the
highest of the pairs being first. 2525, 2624, 2723, 2822, 2921, 3020, 3119, 3218, 3317, 3416, 3515, 3614, 3713, 3812, 3911, 4010, 4109, 4208, 4307, 4406, 4505 On the inside cover of your notebook, you have the list of the 21
words—from the book you intend using—followed by the 21 possible totals.
It’s an easy matter to steal a glance at the prepared list as you open
your note book to jot down your written word.
Further investigations: - Can use your imagination to find other tricks using the unique properties of the dice.
- Are there other number which will also work?
Python program to find all of the sums, unique sums and pairs for book trick: __author__ = 'johnsommer'# url for description of the dice and unique properties # http://www.creativecrafthouse.com/index.php?main_page=product_info&products_id=824 d = list(()); # list of Heath's dice d.append([483, 285, 780, 186, 384, 681]); d.append([642, 147, 840, 741, 543, 345]); d.append([558, 855, 657, 459, 954, 756]); d.append([168, 663, 960, 366, 564, 267]); d.append([971, 377, 179, 872, 773, 278]); dice = list(()); totals = list(); totals_sorted = list(); # Find all possible totals of 5 dice when rolling them for i1 in d[0]: for i2 in d[1]: for i3 in d[2]: for i4 in d[3]: for i5 in d[4]: dice.append ([i1,i2,i3,i4,i5, i1+i2+i3+i4+i5]); totals.append (i1+i2+i3+i4+i5); # print(i1,i2,i3,i4,i5, i1+i2+i3+i4+i5); # done totals_sorted = sorted(totals); dice.sort(key=lambda e: e[5]); # get a list of unique sums totals_set = set(totals_sorted); # hilow list is to represent the list of unique sum using the 1st 2 digits and last 2 digits as pairs. # Arrange the ser of pairs with the highest of the pairs first. hilow = list(); for t1 in totals_set: first2 = int(str(t1)[0:2]); last2 = int(str(t1)[2:4]); if (first2 > last2): hl = first2*100 + last2; else: hl = last2*100 + first2; hilow.append (hl); # Create a unique list hilow = set(hilow); # print( len(dice), len(totals), dice[0], totals[0],totals_sorted[0], len(totals_set), len(hilow)); print('5 unique set of dice'); for i in d: print(i); print(len(dice)); print('Total number of possible dice rolls', len(dice)); print('Number of unique sums = ', len(totals_set)); print('List of unique sums = ', sorted(totals_set)); print('Number of unique sums using first 2 digits and last 2 - highest of pair first', len(hilow)); print('Unique sums using first 2 digits and last 2 - highest of pair first', sorted(hilow)); # print(sorted(totals_set)); # print(hilow); |