Henry Ernest Dudeney/Puzzles and Curious Problems/171 - Multiplication Dates/Solution

== by : $171$ ==
;Multiplication Dates
:In the year $1928$ there were $4$ dates which, when written in the form dd/mm/yy,
::the day multiplied by the month equal the year.
:These are 28/1/28, 14/2/28, 7/4/28 and 4/7/28.
:How many times in the $20$th century -- inclusive -- does this so happen?
:Or, you can try to find out which year in the century gives the largest number of dates that comply with the conditions.
:There is one year that beats all the others.
== Solution ==
According to , there are $215$ dates between $1$st January $1901$ and $31$st December $2000$ in such a form.
 includes such dates as $25/4/00$, where he interprets $00$ as $100$.
:$1924$ has the most such dates:
:$24/1/24$
:$12/2/24$
:$8/3/24$
:$6/4/24$
:$4/6/24$
:$3/8/24$
:$2/12/24$
Of course, this applies to any century, not just the $20$th.
== Proof ==
January:
:$1/1/01$
to:
:$31/1/31$
a total of $31$
February:
:$1/2/02$
to:
:$28/2/56$
a total of $28$
March:
:$1/3/03$
to:
:$31/3/93$
a total of $31$
April:
:$1/4/04$
to:
:$25/4/00$
a total of $25$
May:
:$1/5/05$
to:
:$20/5/00$
a total of $20$
June:
:$1/6/06$
to:
:$16/6/96$
a total of $16$
July:
:$1/7/07$
to:
:$14/7/98$
a total of $14$
August:
:$1/8/08$
to:
:$12/9/96$
a total of $12$
September:
:$1/9/09$
to:
:$11/9/99$
a total of $11$
October:
:$1/10/10$
to:
:$10/10/00$
a total of $10$
November:
:$1/11/11$
to:
:$9/11/99$
a total of $9$
Decmber:
:$1/12/12$
to:
:$8/12/96$
a total of $8$
Hence there are:
:$31 + 28 + 31 + 25 + 20 + 16 + 14 + 12 + 11 + 10 + 9 + 8 = 215$
These can be enumerated:
:$1/1/01$
:$2/1/02$
:$1/2/02$
:$1/3/03$
:$3/1/03$
:$4/1/04$
:$2/2/04$
:$1/4/04$
:$5/1/05$
:$1/5/05$
:$6/1/06$
:$3/2/06$
:$2/3/06$
:$1/6/06$
:$7/1/07$
:$1/7/07$
:$8/1/08$
:$4/2/08$
:$2/4/08$
:$1/8/08$
:$9/1/09$
:$3/3/09$
:$1/9/09$
:$10/1/10$
:$5/2/10$
:$2/5/10$
:$1/10/10$
:$11/1/11$
:$1/11/11$
:$12/1/12$
:$6/2/12$
:$3/4/12$
:$4/3/12$
:$2/6/12$
:$1/12/12$
:$13/1/13$
:$14/1/14$
:$7/2/14$
:$2/7/14$
:$15/1/15$
:$5/3/15$
:$3/5/15$
:$16/1/16$
:$8/2/16$
:$4/4/16$
:$2/8/16$
:$17/1/17$
:$18/1/18$
:$9/2/18$
:$6/3/18$
:$3/6/18$
:$2/9/18$
:$19/1/19$
:$20/1/20$
:$10/2/20$
:$5/4/20$
:$4/5/20$
:$2/10/20$
:$21/1/21$
:$7/3/21$
:$3/7/21$
:$22/1/22$
:$11/2/22$
:$2/11/22$
:$23/1/23$
:$24/1/24$
:$12/2/24$
:$8/3/24$
:$6/4/24$
:$4/6/24$
:$3/8/24$
:$2/12/24$
:$25/1/25$
:$5/5/25$
:$26/1/26$
:$13/2/26$
:$27/1/27$
:$9/3/27$
:$3/9/27$
:$28/1/28$
:$14/2/28$
:$7/4/28$
:$4/7/28$
:$29/1/29$
:$30/1/30$
:$15/2/30$
:$10/3/30$
:$6/5/30$
:$5/6/30$
:$3/10/30$
:$31/1/31$
:$16/2/32$
:$8/4/32$
:$4/8/32$
:$11/3/33$
:$3/11/33$
:$17/2/34$
:$7/5/35$
:$5/7/35$
:$18/2/36$
:$12/3/36$
:$9/4/36$
:$6/6/36$
:$4/9/36$
:$3/12/36$
:$19/2/38$
:$13/3/39$
:$20/2/40$
:$10/4/40$
:$8/5/40$
:$5/8/40$
:$4/10/40$
:$21/2/42$
:$14/3/42$
:$7/6/42$
:$6/7/42$
:$11/4/44$
:$22/2/44$
:$4/11/44$
:$15/3/45$
:$9/5/45$
:$5/9/45$
:$23/2/46$
:$24/2/48$
:$16/3/48$
:$12/4/48$
:$8/6/48$
:$6/8/48$
:$4/12/48$
:$7/7/49$
:$25/2/50$
:$10/5/50$
:$5/10/50$
:$17/3/51$
:$26/2/52$
:$13/4/52$
:$27/2/54$
:$18/3/54$
:$9/6/54$
:$6/9/54$
:$11/5/55$
:$5/11/55$
:$28/2/56$
:$14/4/56$
:$8/7/56$
:$7/8/56$
:$19/3/57$
:$20/3/60$
:$15/4/60$
:$12/5/60$
:$10/6/60$
:$6/10/60$
:$5/12/60$
:$21/3/63$
:$9/7/63$
:$7/9/63$
:$16/4/64$
:$8/8/64$
:$13/5/65$
:$11/6/66$
:$22/3/66$
:$6/11/66$
:$17/4/68$
:$23/3/69$
:$14/5/70$
:$10/7/70$
:$7/10/70$
:$24/3/72$
:$18/4/72$
:$12/6/72$
:$9/8/72$
:$8/9/72$
:$6/12/72$
:$25/3/75$
:$15/5/75$
:$19/4/76$
:$11/7/77$
:$7/11/77$
:$26/3/78$
:$13/6/78$
:$20/4/80$
:$16/5/80$
:$10/8/80$
:$8/10/80$
:$27/3/81$
:$9/9/81$
:$28/3/84$
:$21/4/84$
:$14/6/84$
:$12/7/84$
:$7/12/84$
:$17/5/85$
:$29/3/87$
:$22/4/88$
:$11/8/88$
:$8/11/88$
:$30/3/90$
:$18/5/90$
:$15/6/90$
:$10/9/90$
:$9/10/90$
:$13/7/91$
:$23/4/92$
:$31/3/93$
:$19/5/95$
:$24/4/96$
:$16/6/96$
:$12/8/96$
:$8/12/96$
:$14/7/98$
:$11/9/99$
:$9/11/99$
:$25/4/00$
:$20/5/00$
:$10/10/00$
Of all these years, it is seen that $24$ has $7$ such dates, as enumerated.
== Sources ==
* : Solutions: $171$. -- Multiplication Dates
* : Answers: $236$. Multiplication Dates
Category:Henry Ernest Dudeney/Puzzles and Curious Problems/Solutions