Problems & Puzzles: Puzzles

Puzzle 1150 Find the following numbers... Two puzzles from Shyam Sunder Gupta, in his recent & interesting book "Creative Puzzles to Ignite Your Mind" Q1. Largest Product of Two Numbers. Find two numbers collectively using each of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 only once, such that the product of the two numbers is the largest possible. Q2. Largest Product from Distinct Numbers Find the largest product which can be obtained from distinct positive integers whose sum is 100.

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During the week 11-17 nov 2023 contributions came from Giorgos Kalogeropoulos, JM Rebert, G. Gusev, M. Branicky, A. Rocheli, E, Vantieghem, K. Wilke

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Giorgos wrote:

Q1.  87531 * 9642 = 843973902

 

Q2. 2 + 3 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 = 100

      2 * 3 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 = 21794572800

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Rebert wrote:

Q1. I found:

 9642 * 87531 = 843973902.

 

 Q2. I found:

 49 * 51 = 2499 and 49 + 51 = 100.

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Gennady wrote:

Q1.  9642 * 87531 = 843973902

Q2.  2 * 3 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 = 21794572800 and 2 + 3 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 = 100

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MIchael wrote:

Q1:  843973902 = 9642 * 87531

Q2:  21794572800 = 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 3 * 2

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Allain wrote:

Q1: 9642 * 87531 = 843973902

Q2: 51 * 49 = 2499

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Emmanuel wrote:

Q1. Biggest product : 843973902 = 87531 * 9642

Q2. Biggest product : 21794572800 = 2*3*5*6*7*8*9*10*11*12*13*14

Ken wrote:

Two puzzles from Shyam Sunder Gupta, in his recent & interesting book "Creative Puzzles
to Ignite Your Mind"
Q1. Largest Product of Two Numbers.
Find two numbers collectively using each of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9
only once, such that the product of the two numbers is the largest possible.
Q2. Largest Product from Distinct Numbers
Find the largest product which can be obtained from distinct positive integers
whose sum is 100.
Solution for Q1: Consider the identity
4xy = (x + y)^2 + (x - y)^2. (1)
This identity implies that the product of two numbers x and y is minimal when the quantity
x – y is minimal. Since x and y are distinct integers, we start with x = 9 and y = 8. Then we
construct new integers x and y by annexing new integers on the right so that the new
integers x and y minimize their difference x – y; e.g. annexing 7 and 6 so that our new
integers minimize the difference x – y, our new integers are are x = 96 and y = 87.
Continuing in this fashion, we are led to x = 9642 and y = 8753. Finally we annex the digit
1 to the smaller product. And the maximal product is 87531*9642 = 843973902.
Solution for Q2 Consider the n numbers a1, a2, …,an .Then applying the
arithmetic-geometric mean inequality to these numbers, we have
(a1 + a2 + + an)/n >= (a1 * a2 * … * an)^(1/n) (2)
with equality only when a1 = a2 = … = an.
Now let a1, a2, …,an be distinct integers such that
a1< a2< ,,, <an and (a1 + a2 + + an) = 100. Then by (2)
(100/n) >= (a1 * a2 * … * an)^(1/n) and the product (a1 * a2 * … * an)is maximized
when each term in the product is equal. This leads to a strategy of choosing the integers ai
to be as close to the arithmetic mean (100/n) as possible and still observing the condition
(a1 + a2 + + an) = 100. Selecting the first integers ai <= [100/n] where [ ] is the
greatest integer function severely limits the choice of second integers aj which are >[100/n]
and still have (a1 + a2 + + an) = 100.
For n=2, 100/2 = 50 and the maximal product is 49*51 =2499 which agrees with the
result obtained using (1) above.
For n=3, 100/3 = 33.3333 … and the maximal product is 32 * 33* 35 =36960.

For n =4, 100/4 = 25 and the maximal product is 23*24*26*27 = 387504.
For n =5, 100/5 = 20 and the maximal product is 18*19*20*21*22=7 = 3160080
For n =6, 100/6 = 16.666…and the maximal product is 14*15*16*17*18*20 = 20563200
.
For n =7, 100/7 = 15.142857…and the maximal product is 11*12*13*14*15*17*18=
110270160.
For n =8, 100/8 = 12.5 and the maximal product is 9*10*11*12*13*14*15*16 =
518918400.
For n =9, 100/9 = 11.111… and the maximal product is 7*8*9*10*11*12*13*14*16 =
1937295360
For n =10, 100/10 = 10 and the maximal product is 5*6*7*8*9*11*12*13*14*15=
5448643200.

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