The trick was invented by Maurice Dagbert, a calculating prodigy born in 1913. Dropping out of school at age 11, he began playing with numbers and devising procedures to support the calculation feats he began doing on stage at age 16. Audience members would shout out mathematical problems, which Dagbert would try to solve quickly. In one performance, he correctly answered the interesting question: "Suppose that an hour contains 37.5 minutes and a minute has 96 seconds. How many seconds old would a person aged 24 years be, figuring six leap years?"
3,478 x 5,685 = ?
While a WWII prisoner, Dagbert perfected his techniques and stage performance; after the war he became a professional mental calculator in his own traveling show. For example, observers reported Dagbert (age 32 and still ignorant of algebra) gave instantaneous answers to these problems:
27 raised to the 3rd power = ?
34 raised to the 4th power = ?
31 raised to the 5rd power = ?
72 raised to the 6th power = ?
99 raised to the 7th power = ?
and solve these problems within the stated times:
cube root of 260,917,119 = ? (4 seconds)
29 raised to the 6th power = ? (13 seconds)
89 raised to the 6th power = ? (10 seconds)
cube root of 49,633,171,875 = ? (50 seconds)
Number of seconds in 58 years = ? (23 seconds)
Dagbert combined these numerical feats with his second love--playing a violin. In one demonstration, he extracted twenty cube roots (each involving 3-digit answers) and multiplied two 5-digit numbers while playing a fantasy from Verdi's opera Il Trovatore. The fantasy took seven minutes, after which Dagbert reportedly laid down his violin and reported the 21 correct answers.
This information regarding Dagbert and his calculating feats is taken from Steven Smith's The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies Past and Present (1983). Smith (who lived in Wenatchee, WA) has written a fascinating 374-page book, fileld with stories and calculating tricks. It is amazing to see what individuals can do with numerical calculations if they put their mind to it. Used copies of the book can still be found...and I strongly recommend getting it (and using it).
The majority of the content in the book can be adapted for use in grade 6-12 classrooms in many diffrent ways. For example, Dagbert's mental trick can be used to both do and learn mathematics. The middle school student:
- Gets practice with producing the six permutations of 3 different digits...or explain why there are only six
- Gains practice with column addition and division
- Can explore if the trick works when 0 is allowed a s a digit
- Can explore if the trick works if the 3 digits are not unique (NOTE: Are there still 6 permutations, with the repeated digits written using two different colros?)
- Can investigate why the trick works mathematically in the first place (NOTE: Some algebra skills are needed here.)
I have delayed it long enough...now how does one do the trick? Let's use the Stu's example sum of 2347 to illustrate. Take the thousand's digit (2) and add it to the remaining 3-digit number (347+2=349). Divide this sum by 9 (349/9=38 r 7), and multiply the remainder by 111 (111x7=777). Add this product to previous total (349+777=1126). Finally, remove the thousand's digit and add it to remaining 3-digits (126+1=127). You now know Stu's secret three digits (1,2,7) and the permutation (127) that was added twice. Simple, right! Try it out many times yourself, before trying to perform this feat in public.