#math trivia for #May3: How many day-numbers, like #124, have the property that the no digit is larger than the one’s digit?
— Burt Kaliski Jr. (@modulomathy) May 4, 2012
This is a little different than the answer to #123, where the digits had to be in increasing order. (And to be clear, the “the” in “the no” is an error; it should be “that no digit …”.) The hundreds and the tens digits can be equal to the ones digit, they just can’t be larger.
The possibilities are as follows:
- 1-9: all 9
- 10-99: all of the form xy where y >= x: 9+8+7+6+5+4+3+2+1 = 45
- 100-199: all of the form 1xy where y >= x and y >= 1: 9+9+8+7+6+5+4+3+2+1 = 54
- 200-299: all of the form 2xy where y >= x and y >= 2: 8+8+8+7+6+5+4+3+2+1 = 52
- 300-366: all of the form 3xy where y >= x, y >= 3, and xy <= 66: 7+7+7+7+6+5+1 = 40
The answer is 200. More than half of the day-numbers have the property.