Sunday Times Teaser 2629 – Random Road

by Danny Roth

The 99 houses along Random Road lie on one side only but are not numbered from 1 to 99 in the normal way.

The person who numbered the houses gave the first house a number equal to his age in years and then repeatedly added this number to give consecutive houses their numbers by ignoring multiples of 100 in the numbers produced.

Luckily each house got a different number and George’s number was the same as that he would have had if the houses had been numbered normally.

Luckily each house got a different number and George’s number was the same as that he would have had if the houses had been numbered normally. But he noticed that the house next door had a number like theirs but with the two digits in reverse order.

What is George’s number and how old was the person who put up the numbers?

2 Comments Leave one →

brian gladman permalink

I had Internet problems yesterday so my apologies for the delay in posting.

It turns out that this teaser is ambiguous since the phrase “the next door house had a number like theirs but with the two digits in reverse order” could mean either “a next door house has this house’s number but with its digits reversed”, or “a next door house has its normal number but with its digits reversed”.

The code below finds a solution for each of these interpretations:


fs = "House number is {:d}, numberer's age is {:d}" 
# sensible age range for numberer
for age in range(10, 81):
  # create the house numbers he used as a list
  house_nbrs = [x % 100 for x in range(age, 100 * age, age)]
  # check that all numbers (1 .. 99) are used
  if 0 not in house_nbrs and len(set(house_nbrs)) == 99:
    
    # now find a house whose number and 'normal' number are the same 
    for house_nbr in house_nbrs: 
      normal_nbr = house_nbrs.index(house_nbr) + 1      
      if house_nbr == normal_nbr:
        prv, nxt = (house_nbr - age) % 100, (house_nbr + age) % 100 
        
        # if a next door number has this house's two digits reversed
        prt = 10 * (house_nbr % 10) + house_nbr // 10 in (prv, nxt)
        # if a next door number is its 'digit reversed' normal number
        prt |= (10 * (prv % 10) + prv // 10 == normal_nbr - 1 or
                10 * (nxt % 10) + nxt // 10 == normal_nbr + 1)
        if prt:
          print(fs.format(house_nbr, age))

fs = "House number is {:d}, numberer's age is {:d}"

# sensible age range for numberer

for age in range(10, 81):

# create the house numbers he used as a list

house_nbrs = [x % 100 for x in range(age, 100 * age, age)]

# check that all numbers (1 .. 99) are used

if 0 not in house_nbrs and len(set(house_nbrs)) == 99:

# now find a house whose number and 'normal' number are the same

for house_nbr in house_nbrs:

normal_nbr = house_nbrs.index(house_nbr) + 1

if house_nbr == normal_nbr:

prv, nxt = (house_nbr - age) % 100, (house_nbr + age) % 100

# if a next door number has this house's two digits reversed

prt = 10 * (house_nbr % 10) + house_nbr // 10 in (prv, nxt)

# if a next door number is its 'digit reversed' normal number

prt |= (10 * (prv % 10) + prv // 10 == normal_nbr - 1 or

10 * (nxt % 10) + nxt // 10 == normal_nbr + 1)

if prt:

print(fs.format(house_nbr, age))

Frits permalink

I don’t see much ambiguity otherwise the phrase “had a number like theirs” would have little meaning.


# age range for numberer
for age in range(1, 100):
  # create the house numbers he used as a list
  hns = [x % 100 for x in range(age, 100 * age, age)]

  # check that all numbers (1 .. 99) are used
  if 0 not in hns and len(set(hns)) == 99:
    # now find a house whose number and 'normal' number are the same
    # it should be a 2-digit number that can be reversed to a different number
    # luckily i-boundary 99 will be rejected (no IndexError for hns[i])
    sol = [hn for i, hn in enumerate(hns[9:], 10) if i == hn and
          all(hn % j for j in (10, 11)) and  # must be reversable
          int(str(hn)[::-1]) in {hns[i - 2], hns[i]}]
    for s in sol:
      print(f"House number is {s}, numberer's age is {age}")

# age range for numberer

for age in range(1, 100):

# create the house numbers he used as a list

hns = [x % 100 for x in range(age, 100 * age, age)]

# check that all numbers (1 .. 99) are used

if 0 not in hns and len(set(hns)) == 99:

# now find a house whose number and 'normal' number are the same

# it should be a 2-digit number that can be reversed to a different number

# luckily i-boundary 99 will be rejected (no IndexError for hns[i])

sol = [hn for i, hn in enumerate(hns[9:], 10) if i == hn and

all(hn % j for j in (10, 11)) and # must be reversable

int(str(hn)[::-1]) in {hns[i - 2], hns[i]}]

for s in sol:

print(f"House number is {s}, numberer's age is {age}")

Sunday Times Teaser 2629 – Random Road

by Danny Roth

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