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

Puzzles and Curious Problems by Henry Ernest Dudeney: $171$

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 -- $\text {1901}$ – $\text {2000}$ 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 Dudeney, there are $215$ dates between $1$st January $1901$ and $31$st December $2000$ in such a form.

Dudeney 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:

$31 + 28 + 31 + 25 + 20 + 16 + 14 + 12 + 11 + 10 + 9 + 8 = 215$

$\Box$

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.

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $171$. -- Multiplication Dates
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $236$. Multiplication Dates