Considering a positive whole number X which contains more than one digit, let us define R(X) as the number obtained by reversing the digits of X. Neither X nor R(X) can contain any leading zeroes.

# A Conference commenced on a given day precisely at M*X/ 143 minutes past P o'clock ( but before P+1 o'clock) and concluded at M*R(X) / 143 minutes past Q o'clock ( but before Q+1 o'clock) on the same day ; where 11 >=Q > P >=1 with the proviso that P and Q are whole numbers and M is a positive integer greater than 1.

#It was observed that the hour hand and the minute hand had exchanged places during the respective times of commencement and conclusion of the said conference.

# Determine the total number of distinct choices of the pentuplet (M,X,R(X),P,Q) satisfying conditions of the problem.

NOTE:

(i)Any two choices of the pentuplet are defined to be distinct if they differ in the magnitudes of at least one of the five parameters (viz. M,X.R(X),P and Q).

(ii)It may be noted that Q is always greater than P. For example, (P=2,Q=3) may correspond to valid values for P and Q, but (P=3,Q=2) is not feasible.

(In reply to Computer solution by Mindrod)

I had the computer run through a variety of values:

M: 1-1000

X: 11:1001

of the 49M+ pentuplets thus generated, only about 1.3M were feasible, showing that I could have been smarter about generating pentuplets and/or that I was starting to exhaust the space of feasible pentuplets.

Here's what the computer found (start, stop, p, q, m, x, r)

1:35.69   7:7.93  1  7  63  81  18
2:25.98   5:12.13  2  5  5  743  347
3:46.60   9:18.91  3  9  8  833  338
6:47.83   9:33.99  6  9  10  684  486
6:47.83   9:33.99  6  9  20  342  243

Here's how the search ran (feasible/total)

67586 / 1900000
91372 / 2800000
309812 / 11600000
655686 / 24200000
679472 / 25100000
897912 / 33900000
1243786 / 46500000
1267572 / 47400000
1293820 / 49060000

Which shows that toward the end, only about 1.5% of the trials were feasible.

Posted by Bob Smith on 2006-01-24 13:36:11