Herbert de Grote

Baron Herbert de Grote was born in Mexico in 1892. He is best known for his ability to extract roots from huge numbers.

The 17th edition of The Guinness Book of Records, published October 1970, includes the following entry under 'IQ'.

Herbert de Crote [sic] of Baja California, Mexico has been attested to have extracted the 7th root of a 63-digit number by exclusively mental process in 20 minutes in a test in Chicago, Illinois on 16 Oct. 1969. His answer was 858,312,524. No comparable feat has been recorded.

Other roots extracted by de Grote, all of them exact, included the 13th root from a 100-digit number, the 19th root from a 133-digit number, the 23rd root from a 200-digit number, the 37th root from a 300-digit number, and the 73rd root from a 500-digit number.

De Grote's feats came to the attention of Dutch calculator Wim Klein, who decided to challenge for the Guinness 'big root' record himself. Between 1970 and 1978, both de Grote and Klein made numerous attempts on the record, 1975 being a particularly busy year (entries from The Guinness Book of Records):

Wim Klein extracted the 23rd root from a 200-digit number in 10 min 30 secs at Lycee Mixte Nationalise, Lyon, France on 5 Mar 1975.

Baron Herbert de Grote (b. 9 July 1892) of Mexico City, Mexico extracted a 9-digit root from a 300-digit number on 15 May 1975 in his 83rd year.

Wim Klein extracted the 13th root from a 100-digit number at the Zeeman Laboratory in Amsterdam on 19 September 1975 producing the answer 47,272,683 in 5 min 22 secs.

The roots being exact meant that de Grote and Klein were able to work from both ends of the proposed number. Klein, for instance, would employ log tables that he had memorised in order to determine the leading digits of the root in the same way that a non-exact root would be extracted. From the terminating digits of the proposed number Klein would infer a limited number of possibilities for the terminating digits of the root. If the degree of the root and the proposed number are both relative prime to 10, that is neither is a multiple of 2 or 5, then there is a one-to-one correspondence between the terminating n digits of the root and the terminating n digits of the proposed number, and this is true whatever the value of n.

Here, for example, is how Klein extracted the 13th root from a hundred-digit number in 5 min 22 secs. Klein was given the following hundred-digit number, which was written on a blackboard.

5887531891962289106235631806290799176916689613969290314968338179449060524940136369398710257657355163

Klein began by factorising 58875 into 3 x 5³ x 157 (Klein excelled at factorising). He then added the logarithms of 3, 125 and 157 (Klein knew the logs to five figures of all integers up to 150 and of all primes below 1000). In this way he would arrive at 99.76993 as the (approximate) logarithm of the hundred-digit number. Dividing this by 13 gave him the logarithm of the 13th root, 7.67461.

Next, Klein recalled the logs of 47 and 48, which are 1.67210 and 1.68124 respectively. Linear interpolation told Klein that the he should be looking for a root close to 47250000. By trial and error, Klein determines that the antilog of 7.67461 lies somewhere between 47270000 and 47280000.  At this point Klein writes '4727' on the blackboard. Klein calculates the log of 4727 (= 29 x 163) as 3.67459 and the log of 4728 (= 2³ x 3 x 197) as 3.67468, indicating that the true antilog of 7.67461 is a little over 47272000.

The final three digits of the hundred-digit number are 163. From this, Klein infers that the final three digits of the 13th root are 683. Klein's initial guess of the root is therefore 47272683. In fact, Klein wrote down the last three digits of the root before deducing the fifth digit. It is in the fifth digit that the main uncertainty lies. There is a limit to how accurately the interpolating can be carried out and a risk that the correct root is actually 47271683, or even 47273683. On this occasion, however, Klein was correct.

In his early attempts on the big root record, Klein would invoke Fermat's little theorem to check his answer. For exact 13th roots, the proposed number and the root must leave the same remainder when divided by 13. So Klein would divide the hundred-digit number by 13 and do the same for his calculated root. If the remainders did not agree, then he would adjust the fifth digit of his root accordingly. He would never expect the fifth digit of his initial estimate of the root to be more than 1 away from the correct digit. If Fermat's little theorem revealed that it was, then it was likely that he had made a mistake elsewhere in the calculation. In later attempts on the record, which were all made against the clock, this check took too much time; a risky answer was to Klein better than a slow answer.

Klein's big root demonstrations were often performed in front of large and expectant audiences, so he would often change the degree of the root in order to ensure a new record each time. Eventually, however, Guinness decided, quite arbitrarily, that extracting the 13th root from a hundred-digit number would be the only big root record to feature in future editions of its book. All that was left for Klein to do - Klein by that time was the only calculator attempting such feats - was to lower the time taken to extract the root. By 1981, Klein had the Guinness record down to 1 min 28.8 secs. Satisfied with this, Klein made no further official attempts on his record, though he did continue giving public demonstrations of big root calculations right up to his death in 1986, sometimes bettering his Guinness record.

The 1991 edition of the Guinness Book of Records included the following entry for the category 'Human Computer':

The fastest extraction of a 13th root from a 100-digit number, was achieved by Jaime Garcia Serrano of Bogota, Colombia in a time of 0.15 sec on 24 May 1989 at the Hilton Hotel, Colombia.

The precise circumstances of the feat are not known, not even to Guinness, but clearly the time of 0.15 seconds is a nonsense. Serrano, a mnemotechnical calculator, probably relied more on his memory skills than on any calculating ability and in so doing exposed the main flaw in this type of record.

The 13th root of a hundred-digit number, where the root is exact, lies between 41246264 and 49238826. There are 7992563 different possible problems, i.e. 7992563 different hundred-digit numbers that have an exact 13th root. This is far too many to memorise. However, there are 800 different possibilities for the first four digits of the root. Assuming the hundred-digit number to be relative prime to ten, i.e. its terminating digit is 1, 3, 7 or 9, which it invariably is in the record books, then there are 4000 different possibilities for the last four digits of the root. Still a lot to memorize, but not beyond the powers of an expert mnemotechnician. Perhaps the most sensible option for the mnemotechnical calculator who simply wants to impress would be to memorise the 400 different possibilities for the last three digits of the root and rely on interpolation and a little guesswork to determine the fifth digit of the root.